•TV 


AN 


ELEMENTARY  TREATISE 


ANALYTICAL   GEOMETRY: 

TRANSLATED  FROM  THE  FRENCH  OF  J.  B.  BIOT, 
FOR  THE  USE  OF  THE 

CADETS  OF  THE  VIRGINIA  MILITARY  INSTITUTE 
AT  LEXINGTON.  VA.: 

AUD  ADAPTED  TO  THE  PRESENT  STATE  OF  MATHEMATICAL  INSTRUCTION  IN  THH 

COLLEGES  OF  THE  UNITED  STATES 

BY 
FRANCIS  H.   SMITH,   A.M., 

IBPERINTENDENT  AND  PROFESSOR  OF  MATHEMATICS  OF  THE  VIRGINIA  MILITARY  INSTITUTE,   LATS 

PROFESSOR  OF  MAI  HEMATICS  IN  HAMPD7.N  SIDNilY'COLLSGi:,  AND  FORMERLY  ASSISTANT 

PROFESSOR  IN  THE  U-MT^D  STATES  MILITARY  ACADEMY  AT  WEST  POIMT. 


Latest  Edition,  Carefully  Reviseo. 


PHILADELPHIA: 

CHARLES  DESILVER: 
GLAXTON,  REMSEN  &  HAFFELFINGER ; 

J.  B.  LIPPINCOTT  &  CO. 

NEW  YORK:  D.  APPLETON  &  CO.    BOSTON:  NICHOLS  &  HALL, 
CINCINNATI:  ROBERT  CLARKE  <t  CO;  WILSON,  IIIXKLE  <t  CO. 

SAN  FRANCISCO:  A.  L.  BANCROFT  &  CO. 

Chicago:  S.  C.  GRIGGS  &  Co.— Charleston,  S.  C.:  J.  M.  GREER  &  SON  :  EDWARD  PERRT 

m.—Baigigh.  N.  C.:  WILLIAMS  &  LAMBETH.— Baltimore,  JA/.:  GvsHiwea 

A  RAfLES;  W.  J.  C.  DILANEY  &  Co.—^'nv  Orleans,  La.:  STEVP.NS  & 

&KYMOCR.— Savannah,  Ga.:  J.  M.  COOPER  &  Cn.—<Vacon.Ga.: 

J.  M.  BOARUMAX. — Augusta.  Ga.:  THOS.  RICHARDS  & 

SOM. — Richmond,  Va. :  WOODHOUSE  &  PAKHAX. 

1874. 


Entered,  according  to  Act  of  Congress,  in  the  year  1871,  by 

CHARLES  DESILVER, 
in  the  Office  of  the  Librarian  of  Congress  at  Washington. 


TO  THE 

BOARD  OF  VISITERS 

OF  THg 

VIRGINIA  MILITARY  INSTITUTE, 

THROUGH    WHOSE    ENCOURAGEMENT    AND    SUPPORT 

THIS  WORK 

HAS    BEEN    UNDERTAKEN; 
AXI>  BT  WHOSE  ZEAL  AND  WISDOM  IN  ORGANIZING  AXD  DIRECTISW 


CAUSE    OP    SCIENCE    HAS    BEEN    PROMOTE!*. 


THE     INTERESTS     OF     THE     STATE     OF     VIRGINIA     ADVANCXP. 


781529 


PREFACE 
TO  THE  FIBST  EDITION, 


THE  original  work  of  M.  BIOT  was  for  many  years 
the  Text  Bouk  in  the  U.  S.  Military  Academy  at  West 
Point.  It  is  justly  regarded  as  the  best  elementary 
treatise  on  Analytical  Geometry  that  has  yet  appeared. 
The  general  system  of  Biot  has  been  strictly  followed. 
A  short  chapter  on  the  principal  Transcendental  Curves 
has  been  added,  in  which  the  generation  of  these 
Curves  and  the  method  of  finding  their  equations  are 
given.  A  Table  of  Trigonometrical  Formula  is  ulso 
appended,  to  aid  the  student  in  the  course  ol  his 
study. 

The  design  of  the  following  pages  has  been  to  pre 
pare  a  Text  Book,  which  may  be  readily  embraced  in 
the  usual  Collegiate  Course,  without  interfering  with 
the  time  devoted  to  other  subjects,  while  at  the  same 
time  they  contain  a  comprehensive  treatise  on  the 
subject  of  which  they  treat 

Virginia  Military  Institute, 
JULY,  1840. 

(iv) 


PREFACE 

TO  THE  SECOND  EDITION, 


THE  application  of  Algebra  to  Geometry  constitutes  ono 
of  the  most  important  discoveries  in  the  history  of  mathe 
matical  science.  Francis  Vieta,  a  native  of  France,  and  one 
of  the  most  illustrious  mathematicians  of  his  age,  was  among 
the  first  to  apply  Geometry  to  the  construction  of  algebraic 
expressions.  He  lived  towards  the  close  of  the  fifteenth  cen 
tury.  The  applications  of  Vieta  were,  however,  confined  to 
problems  of  determinate  geometry;  and  although  greater 
brevity  and  power  were  thus  attained,  no  hint  is  to  be  found 
before  the  time  of  DCS  Cartes,  of  the  general  method  of  repre 
senting  every  curve  by  an  equation  between  two  indetermi 
nate  variables,  and  deducing,  by  the  ordinary  rules  of  algebra, 
all  of  the  properties  of  the  curve  from  its  equation. 

RENE  DES  CARTES  was  born  at  Rennes  in  France  in  1596. 
At  tho  early  age  of  twenty  years,  he  was  distinguished  by 
his  solutions  to  many  geometrical  problems,  which  had  defied 
the  ingenuity  of  the  most  illustrious  mathematicians  of  his 
age. 

Generalizing  a  principle  in  every-day  practice,  by  which 
the  position  of  an  object  is  represented  by  its  distances  from 
others  that  are  known,  Des  Cartes  conceived  the  idea  that  by 
referring  points  in  a  plane  to  two  arbitrary  fixed  lines,  as 
axes,  the  relations  which  would  subsist  between  the  distances 
1*  (v) 


vi  PREFACE. 

of  these  points  from  the  axes  might  be  expressed  by  an  alge 
braic  equation,  which  would  serve  to  define  the  line  connect 
ing  these  points.  If  the  relation  between  these  distances,  to 
which  the  name  of  co-ordinates  was  applied,  be  such,  that 
there  exist  the  equation  x  =  y,  x  and  y  representing  the  co 
ordinates,  it  is  plain  that  this  equation  would  represent  a 
straight  line,  making  an  angle  of  45°  with  the  axis  of  x 
Intimate  as  is  the  connection  between  this  simple  principle 
and  that  applied  in  Geography,  by  which  the  position  of  places 
is  fixed  by  means  of  co-ordinates,  which  are  called  latitude 
and  longitude,  yet  it  is  to  this  conception  that  the  science  of 
Analytical  Geometry  owes  its  origin. 

Having  advanced  thus  far,  Des  Cartes  assumed  the  possi 
bility  of  expressing  every  curve  by  means  of  an  equation, 
which  would  serve  to  define  the  curve  as  perfectly  as  it  could 
be  by  any  conceivable  artifice.  Operating  then  upon  this 
equation  by  the  known  rules  of  algebra,  the  character  of  the 
curve  could  be  ascertained,  and  its  peculiar  properties  de 
veloped.  The  application  of  algebra  to  geometry  would  no 
longer  depend  upon  the  ingenuity  of  the  investigator.  The 
sole  difficulty  would  consist  in  solving  the  equation  represent 
ing  the  curve;  for,  as  soon  as  its  roots  were  obtained,  the 
nature  and  extent  of  the  branches  of  the  curve  would  at  once 
be  known. 

Many  authors  of  deservedly  high  reputation  have  treated 
upon  Analytical  Geometry.  Among  the  most  distinguished 
is  J.  B.  Biot,  the  author  of  the  treatise  of  which  the  following 
is  a  translation. 

The  work  of  M.  Biot  has  more  to  recommend  it  than  the 
mere  style  of  composition,  unexceptionable  as  that  is.  The 
mode  in  which  he  has  presented  the  subject  is  so  peculiar  and 
felicitous,  as  to  have  drawn  from  the  Princeton  Review  the 
high  eulogium  upon  his  work,  of  being  "  the  most  perfect  sci 
entific  gem  to  be  found  in  any  language"  His  discussion 
of  the  Conic  Sections  is  the  finest  specimen  of  mathematical 
reasoning  extant.  He  introduces  his  book,  by  showing  how 


PREFACE.  Tii 

the  positions  of  points  may  be  fixed  and  defined,  first  as 
relates  to  a  plane,  and  then  in  space ;  and  by  a  series  of  ex 
amples,  shows  how  analysis  may  be  applied  to  determine 
solutions  to  various  problems  of  Indeterminate  Geometry 
In  these  discussions,  a  simple  and  general  principle  is  applied 
for  determining  all  kinds  of  intersections,  whether  of  straight 
lines  with  each  other  or  with  curves,  curves  with  curves 
planes  with  each  other  or  with  surfaces,  and,  finally,  of  sur 
faces  with  surfaces.  The  principle  is  simple,  inasmuch  as  it 
involves  nothing  more  than  elimination  between  the  equations 
of  the  lines,  curves,  or  surfaces  which  are  considered ;  and 
it  is  general,  since  it  is  applied  to  every  kind  of  intersection. 
In  discussing  the  Conic  Sections,  two  methods  suggested  them 
selves.  Shall  their  equations  be  obtained  by  assuming  a 
property  of  each  section;  or,  from  the  fact  of  their  common 
generation,  shall  the  principle  previously  established,  for  deter 
mining  any  intersection,  be  applied  to  deduce  their  general 
equation  ?  Most  authors  adopt  the  former  method,  which, 
though  apparently  more  simple,  tends  really  to  obscure  the 
discussion,  since  it  assumes  a  property  not  known  to  belong 
to  a  Conic  Section ;  and  if  this  be  afterwards  proved,  the 
proof  is  postponed  too  long  to  enable  the  student  to  realize, 
while  he  is  studying  these  curves,  that  they  are  in  fact  sec 
tions  from  a  Cone.  Biot,  on  the  other  hand,  assumes  nothing 
with  regard  to  these  sections.  He  presumes,  from  their  com 
mon  generation,  that  they  must  possess  common  or  similar 
properties,  since,  by  a  simple  variation  in  the  inclination  of 
the  cutting  planes  the  different  classes  of  these  curves  are 
produced. 

And  so  it  is  with  the  student.  If  he  find  that  the  circum 
ference  of  a  circle  has  all  of  its  points  equally  distant  from 
its  centre,  analogy  leads  him  at  once  to  seek  for  correspond 
ing  properties  in  the  other  sections.  He  finds  in  the  Ellipse 
the  relation  between  the  lines  drawrn  from  the  foci  to  points 
of  the  curve,  and  that  this  relation  rec  uces  to  the  property 
in  the  circle,  when  the  eccentricity  is  zero.  Corresponding 


viii  PREFACE. 

results  are  also  found  in  the  Parabola  and  Hyperbola.  Could 
a  student  anticipate  such  a  connection  between  these  curves, 
.by  following  the  method  of  discussion  usually  adopted  ?  Why 
should  he  examine  the  Hyperbola  any  more  than  the  Cycloid 
for  properties  similar  to  those  deduced  from  the  Circle  ?  They 
are  treated  as  independent  curves,  and  their  equations  are 
found  and  properties  developed,  upon  the  general  principles 
of  analysis,  without  the  slightest  reference  to  their  common 
origin.  Further,  the  purely  analytic  method  adopted  by 
Biot,  prepares  the  mind  for  the  discussion  of  the  general 
equation  of  the  second  degree  in  the  sixth  chapter,  and  that 
of  surfaces  in  the  seventh,  and  certainly  gives  the  student  a 
better  knowledge  of  his  subject  than  any  other. 

This  edition  has  been  most  carefully  revised.  Some  slight 
changes  have  been  made  in  the  mode  of  discussing  one  or 
two  of  the  subjects,  arid  copious  numerical  examples  in  illus 
tration  have  been  added.  The  appendix  also  contains  a  full 
series  of  questions  on  Analytical  Geometry,  which  it  is  be- 
ieved  will  be  of  great  service  to  the  student. 

Virginia  Military  Institute, 
AUGUST,  1846. 


CONTENTS. 


CHAPTER  I. 

PRELIMINARY    OBSERVATIONS. 

Unit  of  Measure *  Page  1 3 

Construction  of  Equations 14 

Construction  of   Equations  of  the 

second  decree 20 

Signification  of  Negative  Results.  .  23 

Examples 24 

CHAPTER  II. 

DETERMINATE    GEOMETRY. 

Analytical  Geometry  defined 25 

Determinate  Geometry 25 

Having  given  the  Inse  and  altitude 
of  a  Triangle,  to  find  the  side  of 

the  Inscribed  Square 25 

To  draw  a  Tangent  to  two  Circles.     27 
To  construct  a  Rectangle,  when  its 
surface  and  the  difference  between 

its  adjacent  sides  are  given 29 

Rules  for  solving  Determinate  Pro 
blems 30 

Examples 31 

CHAPTER  III. 

INDETERMINATE    GEOMETRY. 
Indeterminate  Geometry  defined. . .    33 


Of  Points  and  a  Right  Line  in  a 
Plane. 

Space  defined 35 

Position  of  a  Point  in  a  Plane  de 
termined  35 

Abscissas  and  Ordinates  defined.  . .  35 

Co-ordinate  Axes  defined 35 

Origin  of  Co-ordinates 35 

Equations  of  a  Point 36 

"          «  the  origin 36 

Equation  of  a  Right  Line  referred 

to  Oblique  Axes 41 

Equation  of  a  Right  Line  referred 

to  Rectangular  A xes 44 

General  Equation  of  a  Right  Line.  48 

Distance  between  two  Points 49 

Equation  of  a  Straight  Line  passing 

through  a  given  Point 50 

Equation  of  a  Straight  Line  passing 

through  two  given  Points 51 

Condition  of  two  Lines  being  pa 
rallel  52 

Angle  between  two  Straight  Lines.  53 

Intersection  of  two  Straight  Lines. .  54 

Examples 54 

Of  Points  and  Slraight  Lines  in  Space. 

Determination  of  a  Point  in  Space.  55 

Equations  of  a  Point 56 

Projections  of  a  Point 57 

Distance  between  two  Points 53 

(xi)          . 


Xll 


CONTENTS. 


Projections  of  a  Line 60 

Equations  of  a  Line 61 

Equations  of  a  Curve 63 

Equations  of  a  Line  passing  through 

a  given  Point 63 

Equations  of  a  Line  passing  through 

two  Points 64 

Angle  between  two  given  Lines.  ..  65 

Conditions  of  Perpendicularity  of 

Lines 69 

Conditions  of  Intersection  of  Lines  71 

Of  the  Plane. 

\  Plane  defined 72 

1  Equation  of  a  Plane 73 

Traces  of  a  Plane 74 

General  Equation  of  a  Plane 75 

Equation  of  a  Plane  passing  through 

three  given  Points 78 

Intersection  of  two  Planes 79 

Of  Transformation  of  Co-ordinates. 

Algebraic  and  Transcendental  curves 
defined 80 

•Discussion  of  a  Curve 80 

Formulse  for  passing  from  one  sys 
tem  of  Co-ordinates  to  a  parallel 
one 81 

Formula?  for  passing  from  Rectan 
gular  to  Oblique  Axes 81 

Formula;  for  passing  from  Oblique 
to  Rectangular 82 

Formula)  for  passing  from  Oblique 
to  Oblique 83 

Transformation  in  Space 83 

Of  Polar  Co-ordinates. 

'Polar  Co-ordinates  defined 89 

Equations  of  Transformation 90 

Equations  of  Polar  Co-ordinates  in 

Space 91 


CHAPTER  IV. 
OF    CONIC    SECTIONS. 

onic  Sections  defined 92 

Equation  of  a  Conic  Surface 93 

jreneral  Equation  of  Intersection  of 

Plane  and  Cone 94 

ircle 94 

Ellipse 95 

Parabola 95 

Hyperbola 95 

Of  the  Circle. 

Equation   of   a  Circle    referred    to 
Centre  and  Axes 90 

Equation  of  Circle  referred   to  ex 
tremity  of  Diameter 98 

Equation  of  Circle  referred  to  Axes 
without  the  Circle 99 

Examples 100 

Equation  of  a  Tangent  Line 101 

«         «       Normal       "     102 

Conjugate  Diameters 104 

Polar  Equation  of  Circle 105 

Of  the  Ellipse. 

General  Equation  of  Ellipse 106 

Equation  of  Ellipse  referred  to 

Centre  and  Axes 110 

Parameter Ill 

Equation  of  Ellipse  referred  to  its 

Vertex Ill 

Supplementary  Chords 114 

Foci  of  the  Ellipse 1J6 

Eccentricity 1J6 

Equation  of  Tangent  Line 117 

Construction  of  Tangent  Line.  ...  119 

Conjugate  Diameters 120 

Subtangent 120 

Normal  Line 121 

Subnormal 1 2S 


CONTENTS. 


ziii 


Relation  between  Normal  and  Tan 
gent  Lines 124 

Equation  of  Ellipse  referred  to  Con 
jugate  Diameters 128 

Method  of  finding  Conjugate  Diam 
eters  when  Axes  are  known. . . .  131 

Polar  Equation  of  Ellipse   32 

«         «         «       "       when  the 
Pole  is  at  the  Focus 134 

Values  for  Radius  Vector  used   in 
Astronomy 1 35 

Equation  of  Ellipse  deduced  from 
one  of  its  properties 136 

Measure  of  its  Surface 137 

Of  the  Parabola. 

Equation  of  Parabola 138 

Parameter 139 

Focus  and  Directrix 140 

To  describe  the  Parabola 141 

Equation  of  Tangent  Line 143 

Subtangent 143 

Equation  of  Normal 144 

Subnormal 144 

Construction  of  Tangent  Line.  . ..  145 

Diameter  of  Parabola 146 

Polar  Equation 148 

Polar  Equation  when  the  Pole  is  at 

the  Focus 149 

Measure  of  Surface 150 

Quadrable  Curves 153 

Of  the  Hyperbola. 

General  Equation  of  the  Hyperbola  154 
Equation  of  the  Hyperbola  referred 

to  its  Centre  and  Axes 157 

Equilateral  Hyperbola 158 

Supplementary  Chords 159 

Foci 160 

To  draw  a  Tangent 162 

Relations  between  Axes  and  Con 
jugate  Diameters ]  63 


Asymptotes 164 

To  construct  Asymptotes 1 65 

Equation  of  Hyperbola  referred  to 
Asymptotes 167 

Power  of  the  Hyperbola 168 

To  describe  the  Hyperbola  by  Points  171 
Polar  Equation  of  the  Hyperbola.  .  171 

Formulae  used  in  Astronomy 173 

Common  Equation  of  Conic  Sec 
tions.  .  .  175 


CHAPTER  V. 

DISCUSSION    OF   EQUATIONS. 

Discussion  of  General  Equation  of 

the  Second  Degree 176 

Classification  of  the  Curves -.,  179 

First  Class,  Ba— 4AC  <o« 179 

Application    to    Numerical    Ex 
amples  182 

Examples  for  practice 184 

Particular  Case  of  the  Circle 184 

Second  Class,  B2  —  4AC  =  o 186 

Examples 187 

Third  Class,  B2—  4AC  >  o 190 

Examples 191 

Equilateral  Hyperbola 195 

General  Examples 196 

Centres  of  Curves 197 

Diameters  of  Curves. 199 

Asymptotes    to    Curves    of   the 

Second  Degree 203 

Identity  of  these  Curves  with  the 

Conic  Sections 205 

Tangent  and  Polar  Lines  to  the 

Conic  Sections..... 209 

General  Equation  of  the  Tangent.  210 
Properties   of   Poles    and  Polar 

Lines 210 

General  Equation  of  Polar  Line..  211 
Given  the  Pole  to  find  its  Pola? 

Line ..  212 


xir 


CONTENTS. 


' 


Given  the  Polar  Line  to  find  its 
Pole 212 

To  draw  a  Tangent  Line  to  a 
Conic  Section  from  a  given 
point 213 

Intersection  of  Curves 215 

Examples 21 C 

CHAPTER  VI. 

CURVES    OF   THE    HIGHER   ORDERS. 

Discussion    of    the    Lemniscata 

Curve 220 

Cissoid  of  Diodes 226 

.r  Conchoid  of  Nicomodes  227 

Trisection  of  a  given  Angle  230 

Duplication  of  the  Cube 230 

Lemniscata  of  Bernouilli 231 

Semi-cubical  Parabola 232 

Cubical  Parabola 232 

Transcendental  Curves 233 

Logarithmic  Curve 233 

Its  properties 234 

The  Cycloid 235 

Its  properties 236 

Its  varieties  237 

Their  Equations 238 

Epitrochoid 238 

Epicycloid 238 

Hypotrochoid 238 

Hypocycloid 238 

Equations  to  these  Curves 239-40 

Cardioide 240 

_     Spirals 241 

,    Spiral  of  Archimedes 241 

General  Equation  to  Spirals 242 

Hyperbolic  Spiral 242 

Parabolic  Spiral 213 

Logarithmic  Spiral 234 

Lituus 245 

Remarks  ..., 245 

Formulas  for  transition  from 
Polar  to  Rectangular  co-ordi 
nates 246 

Curves  to  discuss 246 


CHAPTER  VII. 

SURFACES    OF   THE    SECOND    ORDEE. 

How  Surfaces  are  divided 248 

Equation  of  Surfaces  of  the  Second 

Order 248 

Equation  of  the  Sphere 248 

Surfaces  with  a  Centre 255 

Principal  Sections 256 

Principal  Axes 256 

The  Ellipsoids 256 

Ellipsoid  of  Revolution 259 

Sphere 260 

Cylinder 260 

The  Hyperboloids 260 

Ilyperboloids  of  Revolution 262 

Cone  263 

Cylinder 262 

Planes 203 

Surfaces  without  a  Centre 264 

Elliptical  Paraboloid 264 

Hyperbolic  Paraboloid 265 

Parabolic  Cylinder  266 

Tangent  Plane  to  Surfaces  of  the 

Second  Order 267 

Tangent  Plane  to  Sm*faces  which 

have  a  Centre 269 

GENERAL  EXAMPLES  ON  ANALYTI 
CAL  GEOMETRY 269 

Formula  for  the  Angle  included 
between  a  Line  and  Plane  given 
by  their  Equations 272 

Formula  for  the  Angle  between 
two  Planes 272 

Formula  for  the  distance  between 
a  given  point  and  Plane 273 

Tangent  line  to  the  Circle  273 

Equation  of  a  Tangent  Plane  to 
the  Sphere 276 

Given  a  pair  of  Conjugate  Diame 
ters  of  an  Ellipse,  to  describe 
the  Curve  by  points 279 

Another  method 280 

Same  for  the  Hyperbola 28G 


CONTENTS. 


Having  Driven  a  pair  of  Conjugate 
Diameters  of  an  ellipse,  to  con 
struct  the  Axes 280 

Principle  of  the  Trammels 281 

Equation  of  the  Right  Line  in 
terms  of  the  perpendicular  from 
the  origin 282 

Magical  Equations  of  the  Tangent 
Line  to  Conic  Sections 282 

Different  forms  of  the  Equation 
of  the  Plane 283 

Construction  of  surfaces  of  the 
Second  Order  from  their  Equa 
tions...  ..  283 


Equation  of  the  Parabola  in  terms 
of  the  Focal  Radius  Vector  and 
Perpendicular  on  the  tangent..  286 
Same  for  Ellipse  and  Hyperbola..  287 
Equations  of  these  Curves  referred 
to    the  Central  Radius  Vector 
and  Perpendicular  on  the  Tan 
gent 287 

Directrices  of  these  Curves 286 

Notes 288 

APPENDIX. 

Trigonometrical  Formula; 291 

Questions  on  Analytical  Geometry  293 


ANALYTICAL  GEOMETRY 


CHAPTER  I. 

PRELIMINARY  OBSERVATIONS. 

1.  ALGEBRA  is  that  branch  of  Mathematics  in  which  quan 
tities  are  represented   by  letters,  and  the  operations  to  be 
performed  upon  them  indicated  by  signs.     It  serves  to  ex 
press  generally  the  relations  which  must  exist  between  the 
known  and  unknown  parts  of  a  problem,  in  order  that  the 
conditions  required  by  this  problem  may  be  fulfilled.     These 
parts  may  be  numbers,  as  in  Arithmetic,  or  lines,  surfaces,  or 
solids,  as  in  Geometry. 

2.  Before  we  can  apply  Algebra  to  the  resolution  of  Geo 
metrical    problems,  we   must   conceive  of  a  magnitude  of 
known  value,  which  may  serve  as  a  term  of  comparison  with 
other  magnitudes  of  the  same  kind.     A  magnitude  which  is 
thus  used,  to  compare  magnitudes  with  each  other,  is  called 
a  unit  of  measure,  and  must  always  be  of  the  same  dimension 
with  the  magnitudes  compared. 

3.  In  Linear  Geometry  the  unit  of  measure  is  a  line,  as  a 
foot,  a  yard,  &c.,  and  the  length  of  any  other  line  is  ex 
pressed  by  the  number  of  these  units,  whether  feet  or  yards, 
which  it  contains 

2  13 


14  ANALYTICAL  GEOMETRY.  [CiiAr.  L 

C.ZTA  Let  CD  and  EF  be  two  lines,  \vhich  we  wish  to 
compare  with  each  other;  AB  the  unit  of  measure. 
The  line  CD  containing  AB  six  times,  and  the  line 
EF  containing  the  same  unit  three  times,  CD  and 
EF  are  evidently  to  each  as  the  numbers  6  and  3. 
4.  In  the  same  manner  we  may  compare  surfaces 
with  surfaces,  and  solids  with  solids,  the  unit  of  measure  for 
surfaces,  being  a  known  square,  and  for  solids  a  known  cube 

5.  We  'may. 'now  readily  conceive   lines  to  be  added  to, 
subtra^e.fj  .from,  or  multiplied  by,  each    other,  since  these 
operations   have  only   to  be  performed   upon   the   numbers 
which  represent  them.     If,  for  example,  we  have  two  lines, 
whose  lengths  are  expressed  numerically  by  a  and  b,  and  it 
were  required  to  find  a  line  whose  length  shall  be  equal  to 
their  sum,  representing  the  required  line  by  x,  we  have  from 

the  condition. 

x  =  a  +  by 

which  enables  us  to  calculate  arithmetically  the  numerical 
value  of  x,  when  a  and  b  are  given.  We  may  thus  deduce 
the  line  itself,  when  we  know  its  ratio  x  to  the  unit  of 
measure. 

6.  But  we  may  also  resolve  the  proposed   question  geo 
metrically,  and  construct  a  line  which  shall  be  equal  to  the 
sum  of  the  two  given  lines.     For,  let  I  represent  the  absolute 
length  of  the  line  which  has  been  chosen  as  the  unit  of  mea 
sure,  and  A,  B,  and  X,  the  absolute  lengths  of  the  given  and 
required  lines.     The  numerical  values  a,  b,  x,  will  express 
the  ratios  of  these  three  lines  to  the  unit  of  measure,  that  is, 

we  have, 

A          ,       B  X 

„  =  _,         6=        .       x  =  T. 


CHAP.  I.]  ANALYTICAL  GEOMETRY.  15 

These  expressions  being  substituted  in  the  place  of  a,  6,  x, 

in  the  equation 

x  =  a  +  b, 

the  common  denominator  I  disappears,  and  we  have 

X  =  A  +  B. 

Hence,  to  obtain  the  required  line, 

draw  the  indefinite  line  AB,  and  lay  •*  _  c         ? 
off  from  A  in  the  direction  AB  the  dis 

tance  AC  equal  to  A,  and  from  C  the  distance  CB  equal  to  B, 
AB  will  be  the  line  sought. 

7.  The  construction  of  an  analytical  expression,  consists 
in  finding  a  geometrical  figure,  whose  parts  shall  bear  the 
same  relation  to  each  other,  respectively,  as  in  the  proposed 
equation. 

8.  The  subtraction  of  lines  is  performed  as  reaaivv  as  their 
addition.     Let  a  be  the  numerical  vame  of  tne  sr^.vcr  of  the 
two  lines,  b  that  of  lue  iess;  ana  x  tne  required  difference, 

we  have, 

x  =•  <•  ,  —  b) 

an  expression  Vhic'p.  enables  us  to  calculate  the  numerical 
value  0^  '.f.  wnen  a  and  b  are  known.  To  construct  this 
value,  substitute  as  before,  for  the  numerical  values  a,  b,  x, 

A  B  X 

the  ratios  —  ,  —  ,  —  ,  of  the  corresponding  lines  to  the  unit  of 

measure  ;  the  common  denominator  Z  disappears,  and  the 
equation  becomes 


which  expresses  the  relation  between  the  absolute  lengths  of 
these  three  lines.     Drawing  the  inde 

finite  line  AC,  and  laying  off  from  A  A  _  p          s         c 
a  distance  AB  equal  to  A,  and  from 


16  ANALYTICAL  GEOMETRY.  [CHAP  L 

B  in  the  direction  BA,  a  distance  BD  equal  to  B,  AD  will 
express  the  difference  between  A  and  B. 

9.  Comparing   this   solution  with    that   of  the  preceding 
question,  we  see  by  the  nature  of  the  operations  themselves, 
that  the  direction  of  the  line  BD  or  B  is  changed;  when  the 
sign  which  affects  the  numerical  value  of  B  is  changed.    This 
analogy  between  the  inversion  in  position  of  lines,  and  the 
changes  of  sign  in  the  letters  which  express  their  numerical 
values,  is  often  met  with  in  the  application  of  Algebra  to 
Geometry,  and  we  shall  have  frequent  occasion  to  verify  it, 
in  the  course  of  this  treatise. 

10,  From  the  combination  of  quantities  by  addition  and 
subtraction,  let  us  pass  to  their  multiplication  and  division. 
Let  us  suppose,  for  example,  that  an  unknown  line  X  depends 
upon  three  given  lines  A,  B,  C,  so  that  there  exists  between 
their  numerical  values  the  following  relation, 

ob 

x  =  — 
c 

This  relation  enables  us  to  calculate  the  value  of  x,  when 
a,  b,  and  c  are  known.  To  make  the  corresponding  geome 
trical  construction,  substitute  for  a,  b,  c,  and  x,  the  ratios 

A    Tl   C*  ~5C 

__,  _,  _,  __,  of  the  corresponding  lines  to  the  unit  of  measure ; 

I  disappears  from  the  fraction,  and  we  have 

\  =  AB 

""C" 

from  which  we  see  that  the  required  line 
is  a  fourth  proportional  to  the  three  lines 
A,  B,  C.  Draw  the  indefinite  lines  MB 
and  MX,  making  any  angle  with  each 
other;  Lay  off  MC  =  C,  MB  =  B,  and  MA  =  A,  join  C  and 


CHAP.  I.]  ANALYTICAL  GEOMETKY.  17 

A,  and  draw  BX  parallel  to  CA,  MX  is  the  required  line 
For,  the  triangles  MAC,  MXB,  being  similar,  we  have 
MC  :  MB  :  :  MA  :  MX 
C  :  B  :  :  A  :  X 

A  R 

and  consequently  X  =  — 

C 

which  fulfils  the  required  conditions.* 

11.  In  the  example  which  we  have  just  discussed,  as  well 
as  in  the  two  preceding,  when  we  have  passed  from  the  nu 
merical  values  of  the  lines,  to  the  relations  between  their  ab 
solute  lengths,  we  have  seen  that  the  unit  of  measure  /  has 
disappeared;  so  that  the  equation  between  the  absolute 
lengths  was  exactly  the  same  as  that  between  the  numerical 
values.  We  could  have  dispensed  with  this  transformation 
in  these  cases,  and  proceeded  at  once  to  the  geometrical  con 
struction,  from  the  equation  in  a,  b,  and  x,  by  considering 
these  letters  as  representing  the  lines  themselves.  But  this 
could  not  be  done  in  general.  For,  this  identity  results  from 
the  circumstance  that  the  proposed  equations  contain  only 
the  ratios  of  the  lines  to  each  other,  independently  of  their 
absolute  ratio  to  the  unit  of  measure.  This  will  be  evident 
if  we  observe  that  the  equations 

x  =  a  +  b,  x  —  a  —  b, x=  — 

c 

may  be  put  under  the  following  forms, 

i       a  ,  b     ,       a        b     -.      ab 
1  =  —  -f-  — ,  1  =  —  —  — ,   1  =  — 

XX  XX  CX 

*  In  this  example,  as  well  as  those  which  follow,  the  large  letters,  A,  B, 
C,  D,  &c.,  are  used  to  express  the  absolute  lengths  of  the  lines ;  and  the 
small  letters,  a,  b,  c,  d,  &c.,  their  numerical  values,  the  ratio  of  the  unit 
of  measure  to  the  lines. 

2*  c 


IB  ANALYTICAL  GEOMETRY.  [CHAP.  1 

which  express  the  ratios  of  a,  b,  c,  and  T,  with  each  other, 
and  whose  form  will  not  be  changed,  if  we  substitute  for  these 

letters  the  equivalent  expressions  — ,  — ,  — ,  — . 

L       L       L       L 

12.  But  it  will  be  otherwise,  should  the  proposed  equation 
besides  containing  the  ratios  of  the  lines  A,  B,  C  and  X,  with 
each  other,  express  the  absolute  ratio  of  any  of  them  to  the 
unit  of  measure.     For  example,  if  we  had  the  equation 

x  —  ab, 

the  numerical  value  of  x  can  be  easily  calculated,  since  it  is 
the  product  of  two  abstract  numbers,  and  this  value  being 
known,  we  can  easily  construct  the  line  which  corresponds 
to  it.  But,  if  we  wished  to  pass  from  this  equation  to  the 
analytical  relation  between  the  absolute  lengths  of  the 
lines  A,  B,  X,  by  substituting  for  #,  b,  x,  the  expressions 

A   B  X 

— ,  — ,  — ,  /  being  of  the  square  power  in  the  denominator  of 

L        L       L 

the  second  member,  and  of  the  first  power  in  the  first  mem 
ber,  it  would  no  longer  disappear,  and  we  should  have,  after 

reducing, 

V_AB 

T' 

in  which  the  line  X  is  a  fourth  proportional  to  the  lines  /,  A, 
B  In  this,  and  all  other  analogous  cases,  we  cannot  suppose 
the  same  relation  to  exist  between  the  absolute  lengths  of  the 
lines  as  between  their  numerical  values  ;  and  this  impossibility 
is  shown  from  the  equation  itself.  For,  if  a,  b,  and  x,  repre 
sented  lines,  and  not  abstract  numbers,  the  product  a  b  would 
represent  a  surface,  which  could  not  be  equal  to  a  line  x. 

13.  By  the  same  principle,  we  may  construct  every  equa 
tion  of  the  form. 


C«AP.  1 1  ANALYTICAL  GEOMETRY. 

a  b  c  d  . 


x  = 


b'  c  d'  ... 

in  which  a,  b,  c,  d,  b',  c',  d',  &c.,  are  the  numerical  values  of 
so  many  given  lines.  If  we  suppose  the  equation  homoge 
neous,  which  will  be  the  case  if  the  numerator  contain  one 
factor  more  than  the  denominator,  then  substituting  for  the 
numerical  values  their  geometrical  ratios,  we  have 

A  B  C  D  .  .  . 
B  C'  1)    ... 

But  the  first  part  — -  may  be  considered  as  representing  a 

line  A",  the  fourth  proportional  to  B',  A,  and  B.     Combining 

C  A"C 

this    line  with    the   following   ratio   — ,    the  product    — _ 

C'  C' 

will  represent  a  new  line  A"',  the  fourth  proportional  to  C', 

A",  and  C.      This  being  combined  with   r  would   give  a 

A"  D 

product  ,   which  may  be  constructed  in  the  same  man 

ner.  The  last  result  will  be  a  line,  which  will  be  the  value 
of  x. 

14.  We  have  supposed  the  numerator  to  contain  one  more 
factor  than  the  denominator.     If  this  had  not  been  the  case, 
I  would  have  remained  in  the  equation  to  make  it  homoge 
neous.     For  example,  take  the  equation 

x  =  a  b  cd 

the  transformed  equation  becomes 

_  ABCD 
~l~ 

an  expression  which  may  be  constructed  in  the  same  manner 
as  the  preceding. 

15.  Besides  the  cases  which  we  have  just  considered,  the 


20  ANALYTICAL  GEOMETRY.  [CHAP.  I 

unknown  quantity  is  often  given  in  terms  of  radical  expres 
sions,  as 


,    x= 

The  first  V  ab,  expresses  a  mean  proportional  between  a 
and  b,  or  between  the  lines  which  these 
values  represent.  Laying  off  on  the  line 
AD,  AB  =  A,  BD  =  B,  and  on  AD  as 
a  diameter  describing  the  semi-circle 
AXD,  BX  perpendicular  to  AB  at  the  point  B,  will  be  the 
value  of  X.  For,  from  the  properties  of  the  circle,  the  line 
BX  is  a  mean  proportional  between  the  segments  of  the 
diameter. 

16.  If  we  take  the  example, 

x=  V  a2  +  62 

it  is  evident  that  the  required  line  is  the  hypothenuse  of  a 
right  angled  triangle,  of  which  the  sides 
are  AB  =  A,  and  BD  =  B  ;  for  we  have 

AD2  =  AB2-f  BD2 

or  X2  =  A2  +  B2 


X  =  V  A2  +  B2 

17.  We  may  also  construct  by  the  right  angled  triangle, 
the  expression 

*  =  v~^—br 

the  required  line  being  no  longer  the  hypothenuse,  but  one  of 
the  sides.     Making  BD  =  A,  and  DA  =  B,  we  have 

AB2  =  AD2  — BD2 
or  X2  =  A2  — B2 


X=  v/A2  — B2 
18.  Let  us  now  apply  these  principles  to  the  example, 


CHAP.  L]  ANALYTICAL  GEOMETRY.  21 


* 


Solving  the  equation  with  respect  to  r,  we  get  the  two 
toots, 

x  =  a  +  V  a2  —  62,       x  =  a  —  ^  a2  —  b\ 

The  radical  part  of  these  expressions  may  be  evidently 
represented  by  a  side  of  a  right  angled  triangle,  of  which  the 
line  A  is  the  hypothenuse,  and  the  line  B  the  other  side. 
Draw  the  indefinite  line 
ZZ' ;  at  any  point  B 
erect  a  perpendicular, 

2        - 

and  make  BC  =  B.  From  x\ 

the  point  C  as  a  centre 
with  a  radius  equal  to  A, 
describe  a  circumference  of  a  circle,  which  will  cut  ZZ', 
generally,  in  two  points  X,  X',  equally  distant  from  B.  The 
segment  BX,  or  BX',  will  represent  the  radical  v/A2  —  B2, 
and  if  from  the  point  B  we  lay  off  on  ZZ',  a  length  BA  =  A, 
A.X=  VA2  —  B2  +  A  will  represent  the  first  value  of  X 
md  AX'  —  A —  \/A2  —  B2  will  represent  the  second  value. 

19.  If  B  =  A,  it  is  evident  that  the  circle  will  not  cut  the 
ine  ZZ',  but  be  tangent  to  it  at  B.     The  two  lines  BX  and 
3X'  will  reduce  to  a  point,  and  AX  and  AX'  will  be  equal  to 
<ach    other,  and   to  the    line   A.     This   result   corresponds 
itrictly  with    the   change  which    the  Algebraic   expression 
jndergoes;   for  a  =  b  causes    the  radical   \/  a2 — b2  to  dis 
appear,  and  reduces  the  second  member  to  the  first  term,  and 
the  two  roots  become  equal  to  a. 

20.  If  B>  A,  the  circle  described  from  the  point  C  as  a 
centre  will  not  meet  the  line  ZZ',  and  the  solution  of  the 
question  is  impossible.     This  is  also  verified  by  the  equation, 


22 


ANALYTICAL  GEOMETRY. 


[CHAP.  L 


for  b^>  a  makes  the  radical   V  a2  —  b*  imaginary,  and  con 
sequently  the  two  roots  are  impossible. 

21.  If  the  second  member  of  the  equation  had  been  posi 
tive,  the  construction  would  have  been  a  little  different.  In 
this  case  we  would  have, 

3,2 ^  ax  =  62  • 

and  the  roots  would  be, 


x  =  a 


b,       x  =  a  —  V  a*  +  b\ 

Here  the  radical  part  is  repre 
sented  by  the  hypothenuse  of  a 
right  angled  triangle,  whose  sides 
are  A  and  B.  Take  DB  =  B ;  at 
the  point  B,  erect  a  perpendicular 
BC  =  A:  DC  will  be  the  radical 
part  common  to  the  two  roots.  If, 

then,  from  the  point  C  as  a  centre,  with  a  radius  CB  =  A, 
we  describe  a  circumference  of  a  circle,  cutting  DC  in  E' 
and  its  prolongation  in  E,  the  line  DE  will  be  equal  tq 
A  +  V  A2  +  B2,  which  will  represent  the  first  value  of  x 
but  the  second  segment  DE'  =  V  A2  -f  B2  —  A  will  onlj 
represent  the  second  root,  by  changing  its  sign,  that  is,  the 
root  will  be  represented  by —  DE'. 

22.  Here  the  change  of  sign  is  not  susceptible  of  anv 
direct  interpretation,  since,  admitting  that  it  implies  an  in 
version  of  position,  we  do  not  see  how  this  happens,  as  there 
is  no  quantity  from  which  DE'  is  to  be  taken.  But  the  diffi 
culty  disappears,  if  we  consider  the  actual  value  of  x  as  a 
particular  case  of  a  more  general  problerr,  in  which  the 
roots  are, 


CHAP.  I.]  ANALYTICAL  GEOMETRY.  23 


x  =  a  +  c  +  V  a2  -f  b\     x  =  a  +  c—  ^  a2  -i-  b2. 

c,  representing  the  numerical  value  of  a  new  line,  which  is 
also  given.  This  form  of  the  roots  would  make  x  depend 
upon  another  equation  of  the  second  degree,  which  \vould  be, 

a*  _  2  (a  +  c)  x  =  b2  —  2  a  c  —  c2; 

in  which,  if  we  make  c  =  o,  we  obtain  the  original  values 
of  x. 

In  the  new  example,  the  construction  of  the  radical  part 
is  precisely  the  same,  for,  taking  DB  =  B  and  BC  =  A,  the 
hypothenuse    DC    will    repre 
sent    V  A2  +  B2.      From    the 
point   C  as    a  centre  with   a 

radius    equal    to    A,   describe  j  J^^  / 

a    circumference   of  a   circle, 


DE   =  A  +  V  A2  +  B2  and 


—  DE'  =  A—  x/AT  +  FT  To 
obtain  the  first  root,  we  have  only  to  add  C  to  DE,  which 
is  done  by  laying  off  DF  =  C,  and  FE  will  represent 
C  +  A  +  V  A2  +  B2.  To  get  the  second  root,  it  is  evident 
DE'  must  be  subtracted  from  DF.  Laying  off  from  D  to  E", 
in  a  contrary  direction,  DE"  =  DE',  FE"  will  be  the  root, 
and  will  be  equal  to  C  4-  A  —  V  A2  +  B2,  and  this  value 
will  be  positive,  if  the  subtraction  is  possible;  that  is  if  C  or 
Its  equal  DF  is  greater  than  DE',  and  negative,  if  less. 

23.  In  general,  when  a  negative  sign  is  attached  to  a 
result  in  Algebra,  it  is  always  the  index  of  subtraction.  If 
the  expresssion  contain  positive  quantities,  on  which  this  sub 
traction  can  be  performed,  the  indication  of  the  sign  rs  satis 
fied.  If  not,  the  sign  remains,  to  indicate  the  operation  yet 


24  ANALYTICAL  GEOMETRY.  [CHAP.  I. 

to  be  performed.  To  interpret  the  result  in  this  case,  we 
must  conceive  a  more  general  question,  which  contains 
quantities,  on  which  the  indicated  operation  may  be  per 
formed,  and  discover  the  signification  to  be  given  to  the 
result. 

EXAMPLES. 

i    r<  abc  +  def —  ghi. 

1.  Construct    L_ J. S — 

/  m 

2.  Construct    V  a. 


3.  Construct    V  or  +  b*  +  c*  4- 


CHAP.  IL]  ANALYTICAL  GEOMETRY.  25 

CHAPTER  II. 

DETERMINATE  GEOMETRY. 

24.  ANALYTICAL  GEOMETRY  is  divided  into  two  parts . 

1st.  Determinate  Geometry,  which  consists  in  the  applica 
tion  of  Algebra  to  determinate  problems,  that  is,  to  problems 
which  admit  of  only  a  finite  number  of  solutions. 

2dly.  Indeterminate  Geometry,  which  consists  in  the  in 
vestigation  of  the  general  properties  of  lines,  surfaces,  and 
solids,  by  means  of  analysis. 

25.  We  will  first  apply  the  principles  explained  in  the  first 
chapter,  to  the  resolution  and   construction  of  problems  of 
Determinate  Geometry. 

Prob.  1.  Having  given  the  base  and  altitude  of  a  triangle, 
it  is  required  to  find  the  side  of  the  in 
scribed  square.     Let  ABC  be  the  pro 
posed  triangle,  of  which  AC  is  the  base, 
and   BH   the   altitude.      Designate  the 
base  by  b,  and  the  altitude  by  //,  and 
let  x  be  the  side  of  the  inscribed  square.    The  side  EF,  being 
parallel  to  AC,  the  triangles  BEF  and  ABC  are  similar;  and 

we  have, 

AC  :  BH  :  :  EF  :  BI, 

or  b  :  h  :  :  x  :  h  —  x. 

Multiplying  the  means  and  the  extremes  together,  and  put 
ting  the  products  equal  to  each  other,  we  have, 
bh  —  bx  =  hx 
bh 


26  ANALYTICAL  GEOMETRY.  [CHAP.  IL 

from  which  the   numerical  value  of  x  may  be  determined, 
when  b  and  h  are  known. 

26.  We  may  also  from  this  expression  find  the  value  of  x 
oy  a  geometrical  construction,  since  it  is  evidently  the  fourth 

proportional  to  the  lines  b  +  h,  b,  and  L 
Produce  AC  to  B',  making  CB'  =  h,  erect 
the  perpendicular  B'H'  =  h,  join  A  and 
H',  and  through  C  draw  CI'  parallel  to 
H'B',  it  will  be  the  side  of  the  required  square,  and  drawing 
through  I'  a  parallel  to  the  base,  DEFG  will  be  the  inscribed 
square.  For,  the  triangles  AB'H',  ACI'  being  similar,  we  have. 

AB'  :  B'H'  :  :  AC  :  CI' 
or  b  +  h  :  h  :  :  b  :  x; 

bh 

hence  a?  =  7— r-r 

b  +  11 

27.  There  are  some  questions  of  a  more  complicated  nature 
than  the  one  which  we  have  just  considered,  but  which  when 
applied  to  analysis  lead  to  the  most  simple  and  satisfactory 
results. 

Prob.  2.  Draw  through  a  given  point  a  straight  line,  so 
that  the  part  intercepted  between  two  given  parallel  lines 
shall  be  of  a  given  length. 

Let  A  be  the  given  point,  BC  and  DE  the  given  parallels 


It  is  required  to  draw  the  line  AI  so  that  the  part  KI  shall 
be  equal  to  C.     Draw  AG  perpendicular  to  DE,  AG  and  FG 


HAP.  II.]  ANALYTICAL  GEOMETRY.  27 

will  be  known ;  and  designating  AG  by  a,  FG  by  b,  and  GI 
by  xt  we  have, 


AI  .  AG  :  :  KI  :  FG 


crc 


or 


AI  :  a  :  :  c  :  6,     hence  AI  =  -T- 


But 

hence       ~  — 


AI  = 


a2  + 


+  x2       and  x  =  ±      V  c2  — 


From  which  we  see  that  the  problem  admits  of  two  solutions, 
but  becomes  impossible  when  b  ]>  c,  that  is,  when  FG  ^>  KI. 
Construction.  —  From  F  as  a  centre,  with  a  radius  equal  to 
J,  describe  the  arc  HH'  ;  GH  will  be  equal  to  V  c2  —  b*, 
and  AI  parallel  to  FH  will  be  the  required  line.  For  the 
similar  triangles  FGH,  AGI,  give 

FG  :  AG  :  :  GH  :  GI, 


or 


V  c2  —  b2  :  x, 


hence  x  =  -7-  vx  c~  —  62. 
o 


The  second  solution  is  given  by  GI'  =  —  GI. 

28.  Prob.  3.  Let  it  be  required  to  draw  a  common  tangent 
to  two  circles,  situated  in  the  same  plane,  their  radii  and  the 
distance  between  their  centres  being  known. 


Let  us  suppose  the  problem  solved,  and  let  MM'  be  the 
common  tangent.  Produce  'MM'  until  it  meets  the  straight 
line  joining  the  centres  at  T.  The  angles  CMT  and  C'M'T 
being  right  the  triangles  CMT  and  C'M'T  will  be  similar 
and  give  the  proportion, 


28  ANALYTICAL  GEOMETRY.  [CHAP.  IL 

CM  :  C'M'  :  :  CT  :  C'T. 

Designating  the  radii  of  the  two  circles  by  r  and  r',  the 
distance  between  the  centres  by  a,  and  the  distance  CT  by 
x,  the  above  proportion  becomes, 

r  :  r'  :  :  x  :  oc  —  a, 
or  rx  —  ra  =  r'x ; 

,  ar 

hence  x  = 7 » 

r  —  r 

which  shows  that  the  distance  CT  —  x  is  a  fourth  propor 
tional  to  the  three  lines  r  —  r',  «,  and  r. 

To  draw  the  tangent  line* 


Through  the  centres  C  and  C',  draw  any  two  parallel 
radii  CN,  C'N',  the  line  NN'  joining  their  extremities  will  cut 
the  line  joining  the  centres,  at  the  same  point  T,  from  which, 
if  a  tangent  be  drawn  to  one  circle,  it  will  be  tangent  to  the 
other  also.  For  the  triangles  CNT,  C'N'T,  will  still  be 
similar,  since  the  angles  at  N  and  N'  are  equal,  and  will  give 
the  same  proportion.  But  to  show  the  agreement  of  this 
construction  with  the  algebraic  expression  for  x,  draw 
through  N',  N'D  parallel  to  CC',  N'D  will  be  equal  to  a,  and 
ND  to  r  —  r';  the  triangles  N'DN,  CNT,  being  similar,  give 
the  proportion, 

ND  :  DN'  :  :  NC  :  CT, 
or  r  — r'  :  a  :  :  r  :  CT; 

hence  CT  = r » 

r  —  r 


CHAP.  II.]  ANALYTICAL  GEOMETRY.  29 

which  is  the  same  value  found  before.  TMM'  drawn  tangent 
to  one  circle,  will  also  be  tangent  to  the  other.  As  two 
tangents  can  be  drawn  from  the  point  T,  the  question  admits 
of  two  solutions. 

29.  If  we  suppose,  in  this  example,  the  radius  r  of  the 
large  circle  to  remain  constant,  as  well  as  the  distance  be 
tween  the  centres,  the  product  ar  will  be  constant.     Let  the 
radius  r  of  the  small  circle  increase,  as  r  increases,  the  de 
nominator  r  —  r'  will  continually  diminish,  and  will  become 

zero,  when  r  =  r'.      The  value  of  x  then  becomes    ^L.  ~ 

o 

infinity.     This  appears  also  from  the  geometrical  construe 
tion,  for  when  the  radii  are  equal,  the  tangent  and  the  line 
joining  the  centres  are  parallel,  and  of  course  can  only  meet 
at  an  infinite  distance. 

If  r'  continue  to  increase,  the  denominator  becomes  nega 
tive,  and  since  the  numerator  is  positive,  the  value  of  x  will 
no  longer  be  infinite,  but  negative,  and  equal  to  —  CT,  which 
shows  that  the  point  T  is  changed  in  position  (Art.  9),  and 
is  now  found  on  the  left  of  the  circle  whose  radius  is  r. 

30.  Prob.  4.   To  construct  a  rectangle,  when  its  surface 
and  the  difference  between  its  adjacent  sides  are  given: 

Let  x  be  the  greater  side,  2a  the  difference,  x  —  2a  will  be 
the  less.  Let  b  be  the  side  of  the  square,  whose  surface  is 
equal  to  that  of  the  rectangle.  This  condition  will  give 

x  (x  —  a)  =  b2  or  x2  —  2<wr  =  62: 
irom  which  we  obtain  the  two  values, 

x  =  a  +  N/~?  +  68,  x  =  a  —  V  a2  +~b* 

These  are  the  same  values  of  x  constructed  in  Art.  18,  the 
3* 


30  ANALYTICAL  GEOMETRY.  [CHAP.  II. 

first  being  represented  by  DE,  the  second  by  —  DE.     But 


we  can  easily  verify  this,  and  show  that  DE  -  a  +  V  a2  -f  b2 
is  the  greater  side  of  the  rectangle.  For,  if  we  subtract  from 
this  value  the  difference  2«,  the  remainder  —  a  +  V  a*  +  b* 
multiplied  by  the  greater  side,  is  equal  to  b2,  the  surface  of  the 
rectangle,  —  a  +  V  a2  +  b2  is  therefore  the  smaller  side. 

31.  We  see  also  that  the  second  value  of  x  taken  with  a 
contrary  sign,  represents  the  smaller  side  of  the  rectangle. 
Hence  the  calculation  not  only  gives  us  the  greater  side, 
which  alone  was  introduced  as  the  unknown  quantity,  but 
also  the  less.  This  arises  from  the  general  nature  of  all 
algebraic  results,  by  virtue  of  which  the  equation  which  ex 
presses  the  conditions  of  the  problem,  gives,  at  the  same 
time,  every  value  of  the  unknown  quantity  which  will  satisfy 
these  conditions.  In  the  example  before  us  we  have  repre 
sented  the  greater  side  by  +  x,  and  have  found  that  its  value 
depended  upon  the  equation 


If  we  !iad  made  the  smaller  side  the  unknown  quantity,  and 
repr-y/ented  its  value  by  —  a1,  which  we  were  at  liberty  to 
do,  ,i  would  have  depended  upon  the  equation 

—  x  (—  x  +  20)  =  b2,  or  x2  —  Zax  =  b\ 

which  is  the  same  equation  as  the  preceding.  Hence,  this 
equation  should  not  only  give  us  the  greater  side,  which  was 
at  first  represented  by  -f  x,  but  also  the  less,  which  in  this 
instance  is  represented  by  —  x. 

32.  The  preceding  examples  are  sufficient  to  indicate  gene 
rally  the  steps  to  be  taken,  to  express  analytically  the  con 
ditions  of  geometrical  problems  : 


HAT.  H.]  ANALYTICAL  GEOMETRY.  31 

1st.  We  commence  by  drawing  a  figure,  which  shall  re 
present  the  several  parts  of  the  problem,  and  then  such  other 
lines,  as  may  from  the  nature  of  the  problem  lead  to  its 
solution. 

2d.  Represent,  as  in  Algebra,  the  known  and  unknown 
parts  by  the  letters  of  the  alphabet. 

3d.  Express  the  relations  which  connect  these  parts  by 
means  of  equations,  and  form  in  this  manner  as  many  equa 
tions  as  unknown  quantities ;  the  resolution  of  these  equations 
will  determine  the  unknown  quantities,  and  resolve  the  pro 
blem  proposed. 


EXAMPLES. 


1.  In  a  right-angled  triangle,  having  given  the  base,  and 
the  difference  between  the  hypothenuse  and  perpendicular; 
find  the  sides. 

2.  Having  given  the  area  of  a  rectangle,  inscribed  in  a 
given  triangle ;  determine  the  sides  of  the  rectangle. 

3.  Determine  a  right-angled    triangle;   having   given  the 
perimeter  and  the  radius  of  the  inscribed  circle. 

4.  Having  given  the  three  sides  of  a  triangle;   find  the 
radius  of  the  inscribed  circle. 

5.  Determine  a  right-angled   triangle,  having  given    the 
hypothenuse  and  the  radius  of  the  inscribed  circle. 

6.  Determine  the  radii  of  the  three  equal  circles,  described 
in  a  given  circle,  which  shall  be  tangent  to  each  other,  and 
also  to  the  circumference  of  the  given  circle. 


32  ANALYTICAL  GEOMETRY.  [CHAP.  11. 

7.  Draw  through  a  given  point  taken  in  a  given  circle,  a 
chord,  so  that  it  may  be  divided  at  the  given  point  into  two 
segments,  which  shall  be  in  the  ratio  of  m  to  n. 

8.  Having  given  two  points  and  a  straight  line;  describe 
a  circle  so  that  its  circumference   shall   pass  through  the 
points  and  be  tangent  to  the  line. 

9.  Draw  through  a  given  point  taken  within  a  circle,  a 
chord  whose  length  shall  be  equal  to  a  given  quantity. 

10.  Having  given  the  radii  of  two  circles,  which  inscribe 
and  circumscribe  a  triangle  whose  altitude  is  knowrn ;  deter 
mine  the  triangle. 

11.  Draw  through  a  given  point  taken  within  a  given  tri 
angle,  a  straight  line  which  shall  bisect  the  triangle. 

12.  Find  the  distance  between  the  centres  of  the  inscribed 
and  circumscribed  circles  to  a  given  triangle. 


CHAP.  Ill]  ANALYTICAL  GEOMETRY.  83 

CHAPTER  III. 

INDETERMINATE  GEOMETRY. 

33.  IN  the  questions  which  we  have  been  considering,  thfj 
conditions  have  limited  the  values  of  the  required  parts. 
We  propose  now  to  discuss  some  questions  of  Indeterminate 
(joometry,  which  admit  of  an  infinite  number  of  solutions. 

For  example,  let  us  consider  any  line  . 

AMM'.  From  the  points  M,  M',  let  fall 
the  perpendiculars  MP,  M'P',  upon  the 
line  AX  taken  in  the  same  plane.  These 

>^ 

perpendiculars  will  have  a  determinate 
length,  which  will  depend  upon  the  nature  and  position  of 
the  line  AMM',  and  the  distance  between  the  points  M,  M', 
&c.  Assuming  any  point  A  on  the  line  AX,  each  length 
AP  will  have  its  corresponding  perpendicular  MP,  and  the 
relation  which  subsists  between  AP,  PM;  AP',  P'M';  for 
the  different  points  of  the  line  AMM'  will  necessarily  deter 
mine  this  line.  Now,  this  relation  may  be  such  as  to  be 
always  expressed  by  an  equation,  from  which  the  values  of 
AP,  AP',  &c.,  can  be  found,  when  those  of  PM,  P'M',  are 
known.  For  example,  suppose  AP  =  PM,  AP'  =  P'M',  &c., 
representing  the  bases  of  these  triangles  by  x,  and  the  per 
pendiculars  by  y,  we  have  the  relation 

In  this  case,  the  series  of  points  M,  M',  &c.,  forms  evidently 
the  straight  line  AMM',  making  an  angle  of  45°  with  AX. 

E 


34  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

34.  Again,   suppose    that    the    condition    established   was 

such,  that  each  of  *he  lines 
PM,  P'M',  should  be  a  mean 
proportional  between  the  dis 
tances  of  the  points  P,  P',  &c., 
from  the  points  A  and  B  taken 

on  the  line  AB.  Calling  PM,  y,  AP,  x,  and  the  distance  AB 
2fl,  we  would  have, 

yz  =  x  (2#  —  x),     or,  yz  =  %ax  —  a;2. 

This  equation  enables  us  to  determine  y  when  x  is  known, 
and  reciprocically,  knowing  the  different  values  of  x,  we  can 
determine  those  of  y.  It  is  evident  that  this  line  is  the  cir 
cumference  of  a  circle  described  on  AB  as  a  diameter. 

35.  The  equations 

y  =  x  and  y*  =  %ax  —  x2 

are  evidently  indeterminate,  since  both  x  and  y  are  unknown. 
If  values  be  given  to  one  of  the  unknown  quantities,  the  cor 
responding  values  of  the  other  may  be  determined.  Such 
equations,  therefore,  lead  to  infinite  solutions.  But  since  we 
can  determine  every  value  of  y  for  every  assumed  value  of  x, 
these  equations  serve  to  determine  all  the  points  of  the  straight 
line  and  circle,  and  may  be  used  to  represent  them. 

36.  Generalizing  this  result,  we  may  regard  every  line  as 
susceptible   of  being   represented   by  an  equation   between 
two  indeterminate  variables ;  arid,  reciprocally,  every  equa 
tion  between  two  indeterminates  may  be  interpreted  geo 
metrically,  and   considered  as  representing  a  line,  the  dif 
ferent  points  of  which  it  enables  us  to  determine.     It  is  this 
more  extended   application    of  Algebra  to  Geometry,   that 
constitutes  the  Science  of  Analytical  Geometry. 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  35 

Of  Points,  and  the  Rigid  Line  in  a  Plane. 

37.  As  all  geometrical  investigations  refer  to  the  positions 
of  points,  our  first  step  must  be  to  show  how  these  positions 
are  expressed  and  fixed  by  means  of  analysis. 

38.  Space  is  indefinite  extension,  in  which  we  conceive  all 
bodies  to  be  situated.    The  absolute  positions  of  bodies  cannot 
be  determined,  but  their  relative  positions  may  be,  by  refer 
ring  them  to  objects  whose  positions  we  suppose  to  be  known. 

39.  The  relative  positions  of  all  the  points  of  a  plane  are 
determined  by  referring  them  to  two  straight  lines,  taken  at 
pleasure,  in  that   plane,  and   making  any  angle  with   each 
other. 

Let  AX  and  AY  be  these  two  lines,  /  , 

every  point  M  situated  in  the  plane  of  T/  J 

these  lines,  is  known,  when  we  know 
its  distances  from  the  lines  AX  and  AY" 
measured  on  the  parallels  PM  and  QM 
to  these  lines,  respectively. 

The  lines  QM,  Q'M',  or  their  equals  AP,  AP',  are  called 
abscissas,  and  the,  lines  PM,  P'M',  or  their  equals  AQ,  AQ', 
ordinates.  The  line  AX  is  called  the  axis  of  abscissas,  or 
simply  the  axis  qfx's,  and  the  line  AY  the  axis  of  ordinates, 
or  the  axis  ofy's.  The  ordinates  and  abscissas  are  designated 
by  the  general  term  co-ordinates.  AX  and  AY  are  then  the 
co-ordinate  axes,  and  their  intersection  A  is  called  the  origin 
of  co-ordinates. 

40.  It  may  be  proper  here  to  remark,  that  the  terms  line 
and  plane  are  used  in  their  most  extensive  signification, — 
that  is,  they  are   supposed  to  extend  indefinitely  in  both 
directions. 


36  ANALYTICAL  GEOMETRY.  [CnAP.  III. 

41.  Let  us  represent  the  abscissas  by  x,  and  the  ordinatea 
by  y,  x  and  y  will  be  variables,*  which  will  have  different 
values  for  the  different  points  which  are  considered.     If,  for 
example,  having  measured  the  lengths  AP,  PM,  which  deter 
mine  the  point  M,  we  find  the  first  equal  to  a,  and  the  second 
equal  to  b,  we  shall  have  for  the  equations  which  fix  this 

point, 

x  =  a,         y  =  b. 

These  are  called  the  equations  of  the  point  M. 

42.  If  the  abscissa  AP  remain  constant,  while  the  ordinate 
PM  diminishes,  the  point  M  will  continually  approach  the 
axis  AX;  and  when  PM  =  o,  the  point  M  will  be  on  this 
axis,  and  its  equations  become 

x  =  a,        y  =  o. 

If  the  ordinate  PM  remain  constant,  while  the  abscissa 
AP  diminishes,  the  point  M  will  continually  approach  the 
axis  AY,  and  will  coincide  with  it  when  AP  =  o  ;  the  equa 

tions  will  then  be, 

x  =  o,         y  =  b. 

finally,  if  AP  and  PM  become  zero  at  the  same  time,  the 
point  M  will  coincide  with  the  point  A,  and  we  have, 


for  the  equations  of  the  origin  of  co  ordinates. 

43.  From  this  discussion  we  see  that,  in  giving  to  the 
variables  x  and  y  every  possible  positive  value,  from  zero  to 

*  Quantities  whose  values  change  in  the  same  calculation  are  called 
variables  ;  those  whose  values  remain  the  same  are  called  constants.  The 
first  letters  of  the  alphabet  are  generally  used  to  designate  constants,  the 
last  letters  variables. 


CHAP.  IU.]  ANALYTICAL  GEOMETRY.  37 

infinity,  \ve  may  express  the  position  of  every  point  in  the 
angle  YAX.  But  how  may  points  situated  in  the  other 
angles  of  the  co-ordinate  axes  be  expressed  I 

Instead  of  taking  YA  for  the 
axis  of  y,  take  another  line,  Y'A', 
parallel  to  YA  and  in  the  same 
plane,  at  a  distance  AA'  =  A, 
from  the  old  axis. 

Calling  x'  the   new   abscissas,     £- 
counted   from   the   origin  A',  we 
have  for  the  point  M,  situated  in  the  angle  Y'A'X, 

AP  =  AA'  +  AT, 

x  =  A  +  xf. 

But  if  we  consider  a  point  M'  in  the  angle  Y'A'A,  we 
have, 

AP  =  AA'  — AT'. 

x  =  A  —  x\ 

Hence,  in  order  that  the  same  analytical  expression, 
x  =  A  +  x, 

may  be  applicable  to  points  situated  in  both  these  angles,  we 
must  regard  the  values  of  a?'  as  negative  for  the  angle  AA'Y', 
so  that  the  change  of  sign  corresponds  to  the  change  of  posi 
tion  with  respect  to  the  axis  A'Y'. 

44.  To  confirm  this  consequence,  and  show  more  clearly 
how  the  preceding  formula  can  connect  the  different  points 
in  these  different  angles,  let  us  consider  a  point  on  the  axis 
A'Y'.  For  this  point  we  have  x'  =  o,  and  the  formula 

x  =  A  -f  of 
4 


becomes 


ANALYTICAL  GEOMETRY. 
x=  +  A. 


[CHAP.  III. 


This  is  the  value  of  the  abscissa  AA'  with  respect  to  AX, 
AY.  But  if  we  wish  that  this  equation  suit  points  on  the 
axis  AY,  for  any  point  of  this  axis  x  =  o,  and  the  preceding 
formula  will  give, 

x'  =  —  A, 

which  is  the  same  value  of  the  abscissa  AA'  referred  to  the 
axis  A'Y'.  The  analytical  expression  for  this  abscissa  be 
comes  then  positive  for  the  axis  AY,  and  negative  for  the 
axis  A'Y',  when  we  consider  the  different  points  of  the  plane 
connected  by  the  equation 

x  =  A  +  x'. 

This  result  applies  equally  to  the  negative  values  of  #,  and 
proves  that  they  belong  to  points  situated  on  the  opposite 
side  of  the  axis  AY  to  the  positive  values. 

45.  Moving  the  axis  AX  parallel  to  itself,  and  fixing  the 

new  origin  at  A",  making 
AA"  =  B,  and  calling  y  the 
new  ordinates  counted  from 
A",  we  have  for  the  point  M 

AY  =  AA"  +  A"Y, 
or          y  —  B  +  y, 

and  AY"  =  AA"  —  A'Y", 
or  y  =  B  —  y 


for  the  point  M'.     To  express  points  situated  on  both  sides 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  39 

of  the  axis  A"X"  by  the  same  formula,  we  must  regard  those 
points  corresponding  to  negative  values  of  y1  as  lying  on  the 
opposite  side  of  the  axes  of  A"X"  to  the  positive  values;  and 
as  this  applies  equally  to  the  axes  AX  and  AY,  we  conclude 
that  the  change  of  sign  in  the  variable  y  corresponds  to  the 
change  of  position  of  points  vyith  respect  to  the  axis  of  ab 
scissas. 

46.  From  what  has  been  said,  we  conclude,  that  if  the 
abscissas  of  points  lying  on  the  right  of  the  axis  of  y  be 
assumed  as  positive,  those  of  points  lying  on  the  left  of  this 
axis  will  be  negative;   and  also  if  the  ordinates  of  points 

ying  above  the  axis  of  a?  be  assumed  as  positive,  those  below 
this  axis  will  be  negative.  We  shall  have,  therefore, 

In  the  first  angle,  x  positive  arid  y  positive; 

In  the  second  angle,  x  negative  and  y  positive; 

In  the  third  angle,  x  negative  and  y  negative; 

In  the  fourth  angle,  x  positive  and  y  negative; 
and  the  equations 

x  =  a,        y  =  ft, 

which  determine  the  position  of  a  point  in  the  angle  YAX, 
become  successively, 

x  =  —  0,  y  =  +  b  ; 
x  =  —  a,  y  =  —b; 
x  =  +  a,  y  —  —  b. 

47.  Let  us  resume  the  equations  x  =  a,  y  =  b,  which  de 
termine  the  positions  of  a  point  in  a  plane,  a  and  b  being 
any  quantities  whatever. 


40  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

The  equation  x  —  a  considered  by 
itself,  corresponds  to  every  point  whose 
abscissa  is  equal  to  a.  Take  AP  =  a. 
Every  point  of  the  line  PM  drawn 
parallel  to  AY,  and  extending  inde 
finitely  in  both  directions,  will  satisfy 
this  condition,  x  =  a  is  therefore  the 

equation  of  a  line  drawn  parallel  to  the  axis  of  y,  and  at  a 
distance  from  this  axis  equal  to  a.  In  like  manner  y  =  b  is 
the  equation  of  a  straight  line  parallel  to  the  axis  of  x.  The 
point  M,  which  is  determined  by  the  equations 

x  =  a,        y  —  b, 

is  therefore  found  at  the  intersection  of  two  straight  lines 
drawn  parallel  to  the  co-ordinate  axes.  The  line  whose 
equation  is  x  •-=  a  will  be  on  the  positive  side  of  the  axis  oft/ 
if  a  is  positive,  and  the  reverse  if  a  is  negative.  If  a  =  o,  it 
will  coincide  with  the  axis  of  y,  and  the  equation  of  this  axis 

will  be 

x  =  o. 

The  straight  line  whose  equation  is  y  =  b  will  be  situated 
above  or  below  the  axis  of  x,  according  as  y  is  positive  or 
negative.  When  y  =  o,  it  will  coincide  with  the  axis  of  x, 
and  the  equation  of  this  axis  is  therefore 


Finally,  the  origin  of  co-ordinates  being  at  the  same  time 
on  the  two  axes,  will  be  defined  by  the  equations 


as  we  have  before  found. 


CHAP,  m.]  ANALYTICAL  GEO-AIETRY.  41 

48.  The  method  which  we  have  used  to  express  analyti 
cally  the  position  of  a  point,  may  be  therefore  used  to  de 
signate  a  series  of  points,  situated  on  the  same  straight  line 
parallel  to  either  of  the  co-ordinate  axes.     Generalizing  this 
result,  we  see,  that  if  there  exist  the  same  relation  between 
the  co-ordinates  of  all  the  points  of  any  line  whatever,  the 
equation  in  x  and  y  which  expresses  this  relation,  must  cha 
racterize  the  line.     Reciprocally,  the  equation  being  given, 
the  nature  of  the  line  is  determined,  since  for  every  value  of 
x  or  y  we  may  find  the  corresponding  value  of  the  other  co 
ordinate. 

49.  An  equation  which  expresses  the  relation  which  exists 
between  the  co-ordinates  of  every  point  of  a  line,  is  called  the 
equation  of  that  line. 

Let  it  be  required  to  find  the  equation  of  a  straight  line 
passing  through  the  origin  of 
co-ordinates,  and  making  an 
angle  a  with  the  axis  of  x. 
Let  the  angle  which  the  co 
ordinate  axes  make  with  each 
other  be  called  ,3.  From  any 
point  M  draw  PM  parallel  to  the  axis  of  y,  we  will  have, 

PM  :  AP  :  :  sin  a  :  sin  (-3  —  a) 


hence  =  __     JL.         or  y  =  x       sn  * 

AP        sin  (j3  —  a)  sin  (,3  —  a) 

As  the  same  relation  between  y  and  x  will  exist  for  every 
point  of  the  line  AM,  the  equation 
4*  F 


42  ANALYTICAL  GEOMETRY.  [CHAP.  Ill 

y  =  x  — *}"-?—  (1) 

sin  (/3  — a) 

is  the  equation  of  a  straight  line  referred  to  ollique  axes. 

The  value  of  a  is  the  same  for  every  point  of  the  line  AM, 
but  varies  from  one  line  to  another.  If  we  suppose  a  to 
diminish,  the  line  AM  will  incline  more  and  more  to  the  axis 
of  x,  and  when  a  =  o  coincides  with  this  axis.  In  this  case 
the  analytical  expression  becomes  y  —  o,  which  is  the  same 
equation  for  the  axis  of  x  which  Was  found  before. 

Again,  let  a  increase.  The  line  AM  approaches  the  axis 
AY  and  coincides  with  it  when  a  —  {3.  In  this  case  the  sin 
(j8  —  a)  =  o,  and  the  equation  becomes  x  =  o,  which  is  the 
equation  of  the  axis  of  y. 

If  a  continue  to  increase,  (ft  —  a)  becomes  negative,  and 
the  equation  becomes 

Sin  " 


sin  (j8  —  «) 

and  is  the  equation  of  the  line  AM'.  When  a  =  180°, 
sin  a  =  o,  and  the  line  coincides  with  the  axis  of  x,  and  we 
have  again  y  =  o. 

Finally,   for   a  >  180°   sin   a   is   negative,   as   well    as 
sin  (/3  —  a),  and  the  equation  becomes 


sin  (/3  —  «) 
and  represents  the  line  MAM".     Hence  the  formula 


sin  ((3  —  a) 

is  applicable  to  every  straight  line  drawn  through  the  origin 
of  co-ordinates,  when  referred  to  oblique  axes. 


CHAP.  III.] 


ANALYTICAL  GEOMETRY. 


43 


/ 


50.  Let  us  now  consider  a  line  A'M'  making  the  same 
angle  a  with  the  axis  of  x, 
but  which  does  not  pass 
through  the  origin;  and  as 
its  inclination  to  the  axis  of  a? 
does  not  determine  its  posi 
tion,  suppose  it  cut  the  axis 
of  y  at  a  distance  AA'  from 
the  origin,  equal  to  b.  The 
equation  of  a  line  parallel  to  A'M',  and  passing  through  the 

origin,  will  be 

sin  a 


The  value  of  any  ordinate  PM  will  be  composed  of  the 


, 

part          = 


sin 


^li_!__  and  MN  =  AA'  =  b.     Hence 
sin  (,3  —  a) 


sin  a 
sin  (,3  —  a) 

which  is  the  most  general  equation  of  a  straight  line  con 
sidered  in  a  plane. 

51.  To  find  the  point  in  which  this  line  cuts  the  axis  of  x, 
make  y  =  o,  which  is  the  condition  for  every  point  of  this 
axis ;  and  making  x  —  o,  determines  the  point  in  which  it 
cuts  the  axis  of  y. 

Should  the  line  A'M'  cut  the  axis  of  y  below  the  origin  of 
co-ordinates,  the  value  of  the  new  ordinate  would  be  less 
than  that  of  the  ordinate  of  the  line  passing  through  the 
origin,  by  the  distance  cut  off  on  the  axis  of  y;  hence  we 
have  for  the  equation  of  the  line, 


sin  a 
^ v  —  0 


sin  (/3  —  a) 


44  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

52.  In  this  discussion  we  have  supposed  the  co-ordinate 
axes  to  make  any  angle  /3  with  each  other.     They  are  most 
generally  taken  at  right-angles,  since  it  simplifies  the  calcu 
lation.     If  therefore  (3  =  90° 

sin  (/3  —  a)  =  sin  (90°  —  a;  =-  &?£#.,• 
and  the  equation  (1)  becomes 

sin  a 

y  =  x 1-  0  =  x  tan  a  •  f  b. 

y         cos  a 

Representing  the  tangent  of  a  by  a,  this  equation  becomes 
y  =  ax  +  b,  (2) 

which  is  the  equation  of  a  right  line  referred  to  rectangular 
axes.  In  this  equation  a  represents  the  tangent  of  the  angle 
which  the  line  makes  with  the  axis  of  x,  and  b  the  distance 
from  the  origin  at  which  it  cuts  the  axis  of  y. 

53.  If  the  line  passed  through  the  origin  of  co-ordinates, 
b  is  zero,  and  the  equation  (2)  becomes 

y  =  ax, 

which  is  the  equation  of  a  right  line  passing  through  the 
origin  of  co-ordinates  when  referred  to  rectangular  axes. 

By  making  y  =  o  in  equation  (2)  we  determine  the  point 
in  which  the  line  cuts  the  axis  of  x,  the  abscissa  of  which  is 


it  therefore  meets  this  axis  on  the  left  of  the  axis  of  y,  and 

at   a  distance from  the  origin. 

By  finding  the  value  of  x  in  equation  (2)  we  get 

x  =  —y ,  (3) 

a y       a  v  ' 


CHAP.  Ill] 


ANALYTICAL  GEOMETRY. 


as  a  represents  the  tangent  of  the  angle  a  which  the  line 

makes  with  the  axis  of  x,   —  will  be  the  cotangent  of  a,  or 

a 

the  tangent  of  the  complement  of  a ;  but  the  complement  of 
a  is  the  angle  which  the  line  makes  with  the  axis  of  y; 
hence,  to  find  the  angle  which  a  line  makes  with  the  axis  of 
ordinates,  we  find  the  value  of  x  in  the  equation  of  this  line 
referred  to  rectangular  axes,  and  the  co-efficient  of  y  will  be 
the  tangent  of  this  angle. 

54.  The  equation 

y  =  +  ax  +  b 

representing  a  straight  line  which  cuts  the  axis  of  y  at  a 
distance  +  b  from  the  origin,  and  makes  an  angle  whose 
trigonometrical  tangent  is  +  a 
with  the  axis  of  x,  its  posi 
tion  will  be  as  indicated  by 
the  line  A'M,  the  distance 
AA'  being  equal  to  +  b,  and 
the  angle  ABM  represent-  &/£ 


But  the  position  of  the  line  A'M  will  evidently  vary  with 
the  signs  of  a  and  b,  since  the  angle  a  will  be  acute  for  a 
positive  tangent,  but  obtuse  for  a  nega 
tive  one.  And  the  line  A'M  will  cut 
the  axis  of  y  above  the  axis  of  x  for  a 
positive  value  of  b,  but  below  this  axis 
for  a  negative  value.  We  therefore 
conclude  that  for  the  equation 

y  =  +  ox  —  b 
the  line  has  the  position  A'M  (fig.  1). 


Fig.  1 


46 


ANALYTICAL  GEOMETRY. 


[CHAP.  III. 


Fig.  2. 


\ 


Fig.  3. 


When  we  have 

y  —  —  <v,r  +  6 

it    assumes    the    direction    A'M 
(fig.  2),  and  when 


y  —  —  o#  - —  5 

it  is  situated  as  in  fig.  3. 


M" 


55.  Should  the  line  be 
parallel  to  the  axis  of  x 
(fig.  4),  the  angles  a  =  o 
and  a  =  o,  and  the  equa 
tion  becomes 


for  the  line  A'M',  and 


Fi£-  4-  for  the  line  A"M". 

56.  If  we  put  the  equation  of  the  line  under  the  form 
x  =ay±:b,  then,  for  the  foregoing  reasons,  a  will  be  the  tan 
gent  of  the  angle  the  line  makes  with  the  axis  of  y.  If  the 
line  be  parallel  to  this  axis,  a  becomes  zero,  and  we  have 


CHAP,  m.] 


ANALYTICAL  GEOMETRY. 


J7 


A" 


x  =  -f  b 
for  the  line  on  the  right  of  the 

axis,  and 

x  =  —  b 
for  the  line  on  the  left  of  the 

axis;  because  a  =  GO;  therefore  - 

-  and  -f  -  also  become  equal 
a  a 

to  o,  and  the  line  should  coin 
cide  with  the  axis  of  y.  The 
insufficiency  of  the  text  may 
be  readily  overcome,  and  should  be. 

57.  By  giving  to  the  constants  a  and  b  particular  values, 
so  many  particular  lines  may  be  represented.     When  a  =  1 
and  b  =  1,  the  line  cuts  the  axis  of  y  at  a  unit's  distance 
from  the  origin,  and  makes  an  angle  of  45°  with  the  axis  of 
x.     Since  a  =  tang  a  =  tang  45°  =  1. 

58.  The  most  general  form  of  an  equation  of  the  first 
degree  between  two  variables  is 


Fig.  5. 


Ay  +  Ex  +  C  =  o, 


from  which  we  have 


B 


B  P 

By  making  a  =  —  -r-   and  b  =  —  -r-  this  equation  reduces  to 
-A.  A. 

y  =  ax  +  ft, 

which  is  the  equation  of  a  straight  line  referred  to  rectan 
gular  axes  as  before  found 


EXAMPLES. 

1.  Construct  the  line  whose  equation  is 


48  ANALYTICAL  GEOMETRY.  [CHAP.  m. 

2.  Construct  the  line  whose  equation  is 

%y  =  4x  —  2. 

3.  Construct  the  line  whose  equation  is 

2.7- —  3y  —  l  =  6x  —  y 

4.  Construct  the  line  whose  equation  is 

\y  —  3a?  +  J  =  ia:  +2. 

59.  From  what  precedes  we  may  find  the  analytical  ex 
pression  for  the  distance 
between  two  points,  when 
we  know  their  co-ordinates 
referred  to  rectangular  axes. 
Let  M',  M",  be  the  given 
points ;  draw  M'Q'  parallel 
to  the  axis  of  xt  the  triangle 
M'M"Q'  gives 


M'M"  =  «J  M  Q' 2  +  M"Q' 2. 

Let  x',  y'-9  represent  the  co-ordinates  of  the  point  M',  x",  y' 
those  of  the  point  M"  ;  M'Q  =  x"  —  x  ,  and  M"Q'  =  y"  —  y' , 
and  representing  the  distances  between  the  two  points  by  D, 
we  have 

D  =  V  (x"  —  x')2  +  (if — y')2. 

If  the  point  M'  were  placed  at  the  origin  A,  we  should  have 

x'  =  o        y'  =  o, 
and  the  value  of  D  reduces  to 


D  -  V  x"2  +  y"\ 
which  is  the  expression  for  the  distance  of  a  point  from  the 


CHAP.  III.]  ANALYTICAL  GEOMETRY. 

origin  of  co-ordinates.  This  value  is 
easily  verified,  for  the  triangle  AMP 
being  right-angled  gives 

AM2  =  AP2  +  PM2, 


49 


D  =  V  x"2  +  y"2. 

60.  Let  it  be  required  to  find  the  equation  of  a  straight 
line,  which  shall  pass  through  a  given  point. 

Let  x,  y',  be  the  co-ordinates  of  the  given  point  M.  As 
the  line  is  straight,  its  equation  will  be  of  the  form  (Art.  52) 

y  =  ax  +  b. 

Since  the  required  line  must  pass 
through  the  point  M,  whose  co-or- 
ainates  are  x,  y',  its  equation  must 
be  satisfied  when  x'  and  y'  are  sub 
stituted  for  x  and  y;  hence  we 
have  the  condition 

y'  =  ax'  +  b. 

But,  as  it  is  in  general  impossible  for  a  straight  line  to  pass 
through  a  given  point  M,  and  cut  the  axis  of  y  at  a  required 
point  P,  (the  distance  AP  being  equal  to  &,)  and  make  an 
angle  with  the  axis  of  x,  whose  tangent  shall  be  a,  one  of  the 
quantities  a  or  b  must  be  eliminated.  By  subtracting  the 
second  of  the  above  equations  from  the  first,  this  elimination 
is  effected,  and  we  have 

y-y'  =  a(x-x')  (4) 

for  the  general  equation  of  a  straight  line  passing  through  one 
point.     This  equation  requiring  but  two  conditions  to  be  ful 
filled,  may  be  always  satisfied  by  a  straight  line. 
5  G 


50  ANALYTICAL  GEOMETRY.  [CHAP.  TIL 

61.  If  the  given  point  be  on  the  axis  of  x,  then  y'  =  o  and 
the  equation  (4)  becomes 

y  =  a(x  —  x) 

should  the  point  be  upon  the  axis  of  y,  x'  =  o,  and  we  have 
y  —  y'  —  ax, 
y  =  ax  +  y'. 

In  the  same  manner,  by  giving  particular  values  to  x'  and  y', 
the  equation  of  any  line  passing  through  a  given  point  may 
be  determined. 

EXAMPLES. 

1.  Find  the  equation  of  a  line  which  shall  pass  through  a 
point  whose  co-ordinates  are  x'  =  —  1    y  =  +  2. 

2.  Find  the  equation  of  a  straight  line  which  shall  pass 
through  a  point  on  the  axis  of  x  whose  abscissa  is  equal  to 
—  3. 

62.  Let  us  now  find  the  equation  of  a  straight  line  which 
shall  pass  through  two  given  points. 

Let  x't  y'  be  the  co-ordinates  of  one  of  the  points,  x",  y" 
those  of  the  other.     The  line  being  straight,  its  equation  will 

be  of  the  form 

y  =  ax  -f  b. 

Since  the  line  must  pass  through  the  point  whose  co-ordinates 
are  x',  y',  these  co-ordinates  must  satisfy  the  equation  of  the 

line,  and  we  have 

y'  =  ax'  +  b. 

But  it  also  passes  through  the  point  whose  co-ordinates  are 
x",  y",  and  we  have  the  second  condition, 

y"  =  ax"  +  b. 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  51 

The  line  having  to  fulfil  the  two  conditions  of  passing  through 
the  two  given  points,  the  two  constants  a  and  b  must  be  eli 
minated.  By  subtracting  the  second-equation  from  the  first, 
and  the  third  from  the  second,  we  have 

y  —  y  =  a  (*  —  *')» 

y—  y"  =  a(x'  —  x"), 

and  by  dividing  these  two  last  equations  the  one  by  the 
other,  we  have 


which  is  the  equation  of  a  straight  line  passing  through  two 
given  points,  in  which  x  and  y  are  the  general  co-ordinates 
of  the  line,  and  a?',  y',  and  x",  y",  the  co-ordinates  of  the  two 
points.  The  angle  which  it  makes  with  the  axis  of  a?  has  for 


a  tangent 


It  is  easy  to  show  that  the  above  equation  fulfils  the  required 
conditions;  for,  by  supposing  x'  =  x"  the  line  will  become 
parallel  to  the  axis  of  y,  and  the  value  for  the  tangent  becomes 


the  tangent  being  infinite,  the  angle  which  the  line  makes 
with  x  is  90°. 

If  y'  =  y",  we  have 


which  is  the  condition  of  the  line,  being  parallel  to  x;  since 
the  angle  being  o,  the  tangent  is  o. 


52  ANALYTICAL  GEOMETRY.  [CHAP.  HI 

EXAMPLES. 

1.  Find  the  equation  of  a  line  passing  through  two  points 
he  co-ordinates  of  which  are  x  =  1,  y'  —  2,  x"  =  o  y"  =  1. 

2.  Find  the  equation  of  a  line  which  shall  pass  through  a 
point  on  the  axis  of  x,  the  abscissa  of  which  is  —  2,  and 
another  on  the  axis  of  y,  the  ordinate  of  which  is  +  1,  and 
construct  the  line. 

63.  To  find  the  conditions  necessary  that  a  straight  line 
be  parallel  to  a  given  straight  line. 

Let 

y  =  ax  +  b 

be  the  equation  of  the   given   line,  in  which  a  and  b  are 
known.     That  of  the  required  line  will  be  of  the  form 

y  =  a'x  +  b', 

in  which  a'  and  b'  are  unknown. 

In  order  that  these  lines  should  be  parallel,  it  is  necessary 
that  they  should  make  the  same  angle  with  the  axis  of  x. 
Hence 

OL  ==  Q  t 

and  the  equation  of  the  parallel,  after  substitution,  becomes 
y  =  ax  +  b', 

in  which  b'  is  indeterminate,  since  an  infinite  number  of  lines 
may  be  drawn  parallel  to  a  given  line. 

64.  To  find  the  angle  included  between  two  lines,  given 
by  their  equations. 

Let 

y  =  ax  +  b  be  the  equation  of  the  first  line, 

jj  —  a'x  +  b'  the  equation  of  the  second  line. 


CHAP,  ill.]  ANALYTICAL  GEOMETRY.  53 

The  first  line  makes  with  the  axis  of  x  an  angle,  the  trigo 
nometrical  tangent  of  which  is  a  ;  Y 

the  second,  an  arigle  whose  tan 
gent  is  a.  The  angle  sought  is 
ABC  =  a  —  «,  since  BAX  = 

ACB  +  CBA.     But  we  have  from      S/ 

7 
Trigonometry, 

tang  OL  —  tang  a 

tang  (a  —  a)  =  TJ—  2  --  -  -2  -- 
1  +  tang  a'  tang  a 

Calling  ABC  =  V,  and  putting  for  tang  a  and  tang  a  a  and 
a',  we  have 

,r        «'  —  * 

tang  V  =  v—  -  -  ,  • 
1  -f-  aa 

If  the  lines  be  parallel,  V  =  o  ;  and  the  tang  V  =  o,  which 
gives  a  —  a'  =  o  and  a  =  a',  which  agrees  with  the  condition 
before  established  (Art.  63). 

If  the  lines  be  perpendicular  to  each  other,  V  =  90°  and 

IT  ^  -  a 

tang  V  =  ,  —  ;  --  ;  =  oo, 
1  +  aa 

which  gives 

1  -f  aa'=  o, 

which  is  the  condition  that  two  straight  lines  should  be  per 
pendicular  to  each  other.  If  one  of  the  quantities  a  or  a'  be 
known,  the  other  is  determined  by  this  equation. 

EXAMPLES. 

1.  Find  the  angles  between  the  lines  represented  by  the 
equations 


5* 


54  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

2.  Find  the  angles   between  the  lines 

y  =  ff* 

y=l. 

3.  Find  the  angles  between  the  lines 


y  =  x. 

4.  Find  the  angle  of  intersection  of  two  straight  lines,  the 
tangent  of  the  angle  which  one  makes  with  the  axis  of  x 
being  +  1,  that  of  the  other  —  1. 

Ans.  tang  V  =  oo. 

5.  Find  the  angle  of  intersection  when  a  =  o  a'  =  1. 

65.  To  find  the  intersection  of  two  straight  lines,  given 
by  their  equations. 

Let 

y  =  a'x+  by 

y  =  a'x  +  b', 

be  the  equations  of  the  two  lines.  As  the  point  of  intersec 
tion  is  on  both  of  the  lines,  its  co-ordinates  must  satisfy  at 
the  same  time  the  two  equations.  Combining  them,  we 
shall  deduce  the  values  of  x  and  y  which  correspond  to  the 
point  of  intersection.  We  have  by  elimination, 

b  —  b'  ab'  —  a'b 

x  =  —  -  -  »  y  =  -  —  • 

a  —  a  a  —  a 

When  a  =  a',  these  values  become  infinite.  The  lines  are 
then  parallel,  and  can  only  intersect  at  an  infinite  distance. 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  65 


EXAMPLES. 

1.  Find  the  co-ordinates  of  the  point  of  intersection  of 
two  lines,  whose  equations  are 

y  =  3r  +  1, 

y  =  2x  +  4. 

Ans.  x  =  3,  y  =  10. 

2.  Find  the  co-ordinates  of  the  point  of  intersection  of 
two  lines,  whose  equations  are 

y  —  X  =  0, 

3y  —  2x=  1. 

Ans.  x  =  1,  y  =  1. 

66.  The  method  which  we  have  just  employed  is  genera., 
and  may  be  used  to  determine  the  points  of  intersection  of 
two  curve  lines,  situated  in  the  same  plane,  when  we  know 
their  equations ;   for,  as  these   points  must  be  at  the  same 
time  on  both  curves,  their  co-ordinates  must  satisfy  the  equa 
tions  of  the  curves.     Hence,  combining  these  equations,  the 
values  we  deduce  for  x  and  y  will  be  the  co-ordinates  of  the 
points  of  intersection. 

Of  Points,  and  the  Straight  Line  in  Space. 

67.  A  point  is  determined  in  space,  when  we  know  the 
length  and  direction  of  three  lines,  drawn  through  the  point, 
parallel  to  three  planes,  and  terminated  by  them. 

68.  For  more  simplicity  we  wrill  suppose  three  planes  at  right 
angles  to  each  other,  and  let  them  be  represented  by  Y'AX 


M. 


56  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

z  XAZ,  ZAY.     Suppose 

*^ . ^f  the    Doint  ~tyL  at  a  dis- 

tance    MM'    from    the 
first  plane,  MM"  from 

\-.p ^__        second,  and  MM'"  from 

the  third.  If  we  draw 
through  these  lines  three 
planes  parallel  to  the 
rectangular  planes,  their  intersection  will  give  the  point  M. 
The  rectangular  planes  to  which  points  in  space  are  referred, 
are  called  Co-ordinate  Planes.  They  intersect  each  other  in 
the  lines  AX,  AY,  AZ,  passing  through  the  point  A  and  per 
pendicular  to  each  other.  The  distance  MM'  of  the  point 
M  from  the  plane  YAX  may  be  laid  off  on  the  line  AZ,  and 
is  equal  to  AR.  Likewise  the  distance  MM''  may  be  laid  off  on 
AY,  and  is  AQ.  Finally,  AP  laid  off  on  AX  is  equal  to  MM'". 

69.  The  lines  AX,  AY,  AZ,  on  which  hereafter  the  re 
spective  distances  of  points  from  the  co-ordinate  planes  will 
be  reckoned,  are  called  the  Co-ordinate  Axes,  and  the  point 
A  is  the  Origin. 

70.  Let  us  represent  by  x  the  distances  laid  off  on  the 
first,  which  will  be  the  axis  of  x,  by  y  those  laid  off  on  At/, 
which  will  be  the  axis  of  y,  and  by  z  those  laid  off  on  AZ, 
which  will  be  the  axis  of  %. 

If  then  the  distances  AP,  AQ,  AR,  be  measured  and  found 
equal  to  a,  b,  c,  we  shall  have  to  determine  the  point  M,  the 
three  equations 

x  —  a,     y  =  b,     z  —  c. 

These  are  called  the  Equations  of  the  point  M. 

71.  The  points  M',  M",  M'",  in  which  the  perpendiculars 


CHAP.  HI.]  ANALYLICAL  GEOMETRY.  57 

from  the  point  M  meet  the  co-ordinate  planes,  are  called  the 
Projections  of  the  point  M. 

These  projections  are  determined  from  the  three  equations 
given  above,  for  we  obtain  from  them 

y  =  b,  x  =  a,  which  are  the  equations  of  the  projection  M', 
x  =  a,  z  =  c,      "  "  "         of  the  projection  M", 

z  =  c,  y  =  b,      "  "  "         of  the  projection  M'" ; 

and  we  see  from  the  composition  of  these  equations,  that  two 
projections  being  given,  the  other  follows  necessarily. 

In  the  geometrical  construction  they  may  be  easily  deduced 
from  each  other.  For  example,  M",  M'",  being  given,  draw 
M"'Q,  M"P,  parallel  to  AZ,  and  QM',  PM ,  parallel  respect 
ively  to  AX  and  AY,  M'  will  be  the  third  projection  of  the 
point  M. 

72.  There  results  from  what  has  been  said,  that  all  points 
in  space  being  referred  to  three  rectangular  planes,  the  points 
in  each  of  these   planes  are  naturally  referred  to  the  two 
perpendiculars,  which   are    the   intersections  of  this   plane 
with  the  other  two. 

The  plane  YAX  is  called  the  plane  of  x'st  and  y's,  or 
simply  xy ; 

The  plane  XAZ,  that  of  x's,  and  %'s,  or  xz  ; 

And  the  plane  ZAY,  that  of  z's,  and  y's,  or  zy ; 

The  same  interpretation  is  given  to  negative  ordinates,  as 
we  have  before  explained,  and  the  signs  of  the  co-ordinates 
x,  y,  z,  will  make  known  the  positions  of  points  in  the  eight 
angles  of  the  co-ordinate  planes. 

73.  Let  us  resume  the  equations, 

x  =  a,     y  =  b,     z  =  c; 
a,  bt  c,  being  indeterminate. 


58  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

The  first  x  —  a  considered  by  itself,  belongs  to  every 
point  whose  abscissa  AP  is  equal  to  a.  It  belongs  therefore 
to  the  plane  MM'PM",  supposed  indefinitely  extended  in 
both  directions.  For  every  point  of  this  plane,  as  it  is  pa 
rallel  to  the  plane  ZAY,  satisfies  this  condition.  The  equa 
tion  y  =  b  corresponds  to  every  point  of  the  plane  MM'" 
QM',  drawn  through  the  point  M  parallel  to  ZAX,  and 
finally  z  —  c  corresponds  to  every  point  of  the  plane  MM" 
RM'"  drawn  through  M  parallel  to  the  plane  XAY.  Hence 
the  equations 

x  =  at     y  =  b,     z  =  c, 

show  that  the  point  M  is  situated  at  the  same  time  on  three 
planes  drawn  parallel  respectively  to  the  co-ordinate  planes 
and  at  distances  represented  by  a,  b,  c. 

When  these  distances  are  nothing,  the  equations  become 

x  =  o,     y  =  o,     2  =  0, 

which  are  the  equations  of  the  origin.  The  first  of  these 
x  =  o  corresponds  to  the  plane  ?/z,  the  second  y  =  o  to  the 
plane  xz,  and  the  third  z  =  o  to  the  plane  xy.  Since  for  every 
point  of  these  planes,  these  separate  conditions  exist. 

74.  To  find  the  expression  for  the  distance  between  two 
points  in  space.  Let  M,  M',  be  the  two  points,  the  co-ordi 
nates  of  the  first  being  a/,  y',  z',  those  of  the  second,  x",  y",  z". 
Draw  MQ  parallel  to  the  plane  of  xy,  and  ([united  by  the 
ordinate  M'N',  we  shall  have)  ^-  tru.^'*  «.€««•  L 


or  since  MQ  -  NN', 


MM    = 


CHAP.  III.] 


ANALYTICAL  GEOMETRY. 


Draw  NR  parallel  to  the 
axis  of  x,  we  shall  have 


NN  -  NR  +  N'R . 
But 

IN  1C  —  X  X  f 

and      N'R  =  y"  —  y', 
hence 

\<*  =  (x''—x')*+(y«—y> 

And  we  have  also 

QM'  =  M'N'  —  MN  -  z"  —  z'. 

Substituting  the  values  of  NN'  and  QM',  we  have 

"MM"'2  =  (x"  —  x')2  +  (y"  —  yj  +  (z"  —  z')2, 


or      MM'  =  D  =  V  (x"  —  a;')8  +  (y"  —  y')2  +  (z"  —  z  )2. 

75.  If  one  of  the  points,  as  for  example  that  whose  co-or 
dinates  are  x,  y,  z',  coincide  with  the  origin,  the  preceding 
formula  becomes 

D=  V  x"2  +  y2  +  z"2, 

which  expresses  the  distance  of  a  point  in  space  from  the 
origin   of  co-ordinates.     In    fact, 
the  triangles  MAM',  AM'P  being 
right-angled  at  M'  and  P,  give 


AM4  =  MM  -  +  AM  '', 


AM2  -  z"2  +  y'2  +  x"2, 
as  we  have  just  found. 

We  see  by  this  result,  that  the  square  of  the  diagonal  of  a 
rectangular  parallelopipedon  is  equal  to  the  sum  of  the  squares 
of  its  three  edges. 


60  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

76.  This  last  result  gives  a  relation  between  the  cosines  of 
the  angles  which  any  line  AM  makes  with  the  co-ordinate 
axes.     For,  let  these  three  angles  be  represented  by  X,  Y,  Z  ; 
call  r  the  distance  AM,  in  the  right-angled  triangle  AMM' 

we  have 

MM' =  2,    AMM'  =  MAZ  =  Z. 
Hence 

z  =  r  cos  Z. 

Reasoning  in  the  same  mariner  we  have 
y  =  r  cos  Y, 
x  =  r  cos  X. 

Squaring  these  three  equations  and  adding  them  together  we 

have 

z*  4.  f  4.  £  =  r2  (cos2  X  +  cos2  Y  +  cos2  Z), 

but  x2  +  y2  +  *  =  r2. 

Hence  cos2  X  +  cos2  Y  +  cos2  Z  =  1, 

which  proves,  that  the  sum  of  the  squares  of  the  cosines  of  the 
angles  which  a  straight  line  in  space  makes  with  the  co-ordi 
nate  axes  is  always  equal  to  unity. 

77.  Let  us  now  determine  the  equations  of  a  straight  line 
in  space. 

To  do  this,  we  will  remark  that,  if  a  plane  be  drawn 
through  a  straight  line  in  space,  perpendicular  to  either  of 
the  co-ordinate  planes,  its  intersection  with  this  plane  will  be 
the  projection  of  the  line  on  that  plane.  The  perpendicular 
plane  is  called  the  projecting  plane.  There  are  therefore 
three  projecting  planes,  and  also  three  projections ;  and  as 
each  of  the  projecting  planes  contains  the  given  line  and  one 
of  its  projections,  knowing  two  of  the  projections,  we  may 
draw  two  projecting  planes  whose  intersection  will  determine 


CHAP  HI]  ANALYTICAL  GEOMETRY.  61 

the  line  in  space.  Hence,  two  projections  of  a  line  in  space 
are  sufficient  fo  determine  it. 

As  these  projections  are  straight  lines,  their  equations  will 
be  of  the  form, 

x  =  az  +  «,  for  the  projection  on  the  plane  of  xz, 
y  =  bz  +  /3,  "  "          on  the  plane  of  yz. 

These  equations  fix  the  position  of  the  line  in  space,  since 
they  make  known  the  projecting  planes,  whose  intersection 
determines  the  line. 

If  the  given  line  passed  through  the  origin  of  co-ordinates, 
we  should  have  a  =  o  and  /3  =  o,  and  the  above  equations 

would  become 

x  =  az, 

y  =  bz. 
78.  These  results  are  easily  verified;  for  the  equation 

x  =  az  +  « 

being  independent  of  y,  is  not  only  the  equation  of  the  pro 
jection  of  the  given  line  on  the  plane  of  xz,  but  corresponds 
to  every  point  of  the  projecting  plane  of  the  given  line,  of 
which  this  projection  is  the  trace.  It  is  therefore  the  equa 
tion  of  this  plane. 

Likewise  the  equation 

y  =  bz  +  fi 

being  independent  of  x,  not  only  represents  the  equation  of 
the  projection  of  the  given  line  on  the  plane  of  yz,  but  is  the 
equation  of  the  plane  which  projects  this  line  on  the  plane 
of  yz.  Consequently  the  system  of  equations 

x  =  az  +  a,  =  bz  +  3 


62  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

signifies  that  the  given  line  is  situated  at  the  same  time  on 
both  these  planes.     Hence  they  determine  its  position. 
79.  Eliminating  z  from  these  equations,  we  get, 

r—  a     y—$  o     b  , 

-  =  ^—j  —  >  or  y  —  p  =  —  (x  —  o), 

^ 


wnich  is  the  equation  of  the  projection  of  the  given  line  on 
the  plane  of  yx,  and  also  corresponds  to  the  plane  which 
projects  this  line  on  the  plane  of  xy. 

80.  We  conclude  from  these  remarks  that,  in  general,  two 
equations  are  necessary  to  fix  the  position  of  a  line  in  space, 
and  these  equations  are  those  of  the  two  planes,  whose  inter 
section  determines  the  line.     When  a  line  is  situated  in  one 
of  the  co-ordinate  planes,  its  projections  on  the  other  two  are 

*•  *  ru  jj.  *.   in  the  axes.A   If,  for  example,  it  be  in  the  plane  of  xz,  we 

ifou**   »U«r:  have  for  any  line  of  this  plane, 
.f.  b  =  o,         fi  =  o; 

and  its  equations  become 

y  =  o,         x  =  az  4-  a. 

The  first  shows  that  the  projection  of  the  line  on  the  plane 
of  yz  is  in  the  axis,  and  the  second  is  the  equation  of  its  pro 
jection  on  the  plane  of  xz,  which  is  the  same  as  for  the  line 
itself,  with  which  it  coincides. 

81.  Let  us  resume  the  equations 

x  =  az  +  a,         y  =  bz  +  j3. 

So  long  as  the  quantities,  a,  b,  a,  /3,  are  unknown,  the  posi 
tion  of  the  line  is  undetermined.  If  one  of  them,  a  for  ex 
ample,  be  known,  this  condition  requires  that  the  line  shall 
have  such  a  position  in  space,  that  its  projection  on  the  plane 
of  xz  shall  make  an  angle  with  the  axis  of  z,  the  tangent  of 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  63 

which  is  a.  If  a  be  also  known,  this  projection  must  cut  the 
axis  of  x  at  this  given  distance  from  the  origin,  and  these 
two  conditions  will  limit  the  line  to  a  given  plane. 

If  b  be  known,  a  similar  condition  will  be  required  with 
respect  to  the  angle  which  its  projection  on  the  plane  of  yz 
makes  with  the  axis  of  z  ;  and  finally,  if  all  four  constant? 
be  known,  the  line  is  completely  determined. 

82.  The  determination  of  the  constants   a,  b,  a,  fi,  from 
given   conditions,   and   the   combination  of   the  lines  which 
result  from  them,  lead  to  questions  which  are  analogous  to 
those  we  have  been  considering. 

Before  proceeding  to  their  discussion,  we  will  remark,  that 
the  methods  which  wre  have  just  used,  may  be  applied  to 
curve  as  well  as  straight  lines.  In  fact,  if  we  know  the 
equations  of  the  projections  of  a  curve  on  two  of  the  co 
ordinate  planes,  we  can  for  every  value  of  one  of  the  varia 
bles  x,  y,  or  z,  find  the  corresponding  values  of  the  other  two, 
which  will  determine  points  on  the  curve  in  space. 

83.  The  projection  of  a  curve  on  a  plane  is  the  intersection 
with  this  plane  by  a  cylindrical  surface,  passed  through  the 
curve  perpendicular  to  the  plane. 

If  we  know  the  equations  of  two  of  its  projections,  these 
equations  show  that  the  curve  lies  on  the  surfaces  of  two 
cylinders,  passing  through  these  projections,  and  perpendi 
cular  to  their  planes  respectively.  Hence  their  intersection 
determines  the  curve. 

The  term  Cylinder  is  used  in  its  most  general  sense,  and 
applies  to  any  surface  generated  by  a  right  line  moving  pa 
rallel  to  itself  along  any  curve. 

84.  To  find  the  equations  of  a  straight  line  passing  through 
a  given  point. 


64  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

Let  x ' ,  y',  z',  be  the  co-ordinates  of  the  given  point.  The 
equations  of  the  line  will  be  of  the  form 

>t  Ikin    f<h.«i».««     TU     <jr««l*    f1*"**-     § u >»•«!•     x  =  OZ  +  a, 

But  since  the  line  must  pass  through  the  given  point,  these 
equations  must  be  satisfied  when  x',  y',  and  z'  are  substituted 
for  x,  y,  and  z.  We  have  therefore  the  conditions 

y'  =  bz'  +  ft 
? 

Eliminating^  and  ft  by  subtracting  the  two  last  equa 
tions  from  the  two  first,  we  have 

x  —  x'  =  a  (z  —  z'), 
for  the  equations  of  a  straight  line  passing  through  the  point 

EXAMPLES. 

1.  Find  the  equations  of  a  straight  line  passing  through 
the  point  whose  co-ordinates  are  x'  =  o,  y'  =  o,  z'  —  1. 

2.  Find  the  equations  of  a  straight  line  passing  through 
the  point  whose  co-ordinates  are  x'  =  —  1,  y'  =  o,  z'  =  +  1. 

85,  To  find  the  equations  of  a  right  line  passing  through 
iwo  given  points. 

Let  x',  y',  z',  x",  y",  z",  be  the  co-ordinates  of  these  points. 
The  equations  of  the  required  line  will  be  of  the  form 

x  =  az  -f  « 

y  =  bz  +  ft 

c,  b,  a,  ft  being  unknown.     In  order  that  the  line  pass  through 


CHAP.  Ifl.]  ANALYTICAL  GEOMETRY.  65 

the  point  whose  co-ordinates  are  x',  y'y  z',  it  is  necessary 
that  these  equations  be  satisfied  when  we  substitute  x  ',  y 
and  z',  for  x,  y,  and  z.  Hence 

x'  =  az1  +  «, 
y'  =  bz'  +  p. 

For  the  same  reason,  the  condition  of  its  passing  through 
the  point  whose  co-ordinates  are  x"9  y",  z"t  requires  that  we 

have 

x"  =  az"  +  «, 

y"  =  bz"  +  0. 

These  equations  make  known  a,  b,  a,  /3,  and  substituting 
their  values  in  the  equation  of  the  straight  line,  it  is  deter 
mined.  Operating  upon  these  equations  as  in  Art.  84,  we 

have 

(x  —  X')  =  a(z  —  z'),         (*'  —  X")  =  a(z'  —  z"), 

(y  -  y'  )  =  b  (z  -  z'),         (y'  -  y")  =  &(*'-  z"), 
from  which  wre  get 


The  two  last  equations  are  those  of  the  required  line,  the 
other  two  make  known  the  angles  which  its  projections  on  the 
planes  of  xz  and  yz  make  with  the  axis  of  s. 

EXAMPLES. 

1.  Find  the  equations  of  a  straight  line  passing  through 
the  points,  whose  co-ordinates  are  x'  =  o,  y'  =  o,  z'  =  —  1  ; 
and  x"  =  I,  y"  =  o,  z"  =  o,  and  construct  the  line. 

2.  Find  the  equations  of  the  line  passing  through  the  origin 

6* 


ANALYTICAL  GEOMETRY.  [CHAP.  Ill 

of  co-ordinates,  and  a  point,  the  co-ordinates  of  which  are 

86.  To  find  the  angle  included  between  two  given  lines. 
Let 

x  =  az  +  a  ; 

be  the  equations  of  the  first  line. 


x  =  az 


those  of  the  second. 


We  will  remark  in  the  first  place,  that  in  space,  two  lines 
may  cross  each  other  under  different  angles  without  meeting, 
and  their  inclination  is  measured  in  every  case  by  that  of 
two  lines,  drawn  parallel  respectively  to  the  given  lines 
through  the  same  point. 

Draw  through  the  origin  of  co-ordinates  two  lines  respec 
tively  parallel  to  those  whose  inclination  is  required,  their 
equations  will  be 

x  =  az  ) 

I   for  the  first, 

=  bz  ) 


y 

x  =  a'z 
y=  b'z 


for  the  second 


Take  on  the  first  any  point  at  a  distance  r  from  the  origin, 
the  co-ordinates  of  this  point  being  x,  yf,  z' ;  and  on  the 
second  line  take  another  point  at  a  distance  r"  from  the  origin, 

and  call  the  co-ordinates  of  this 
point  x",  y",  z",  and  let  D  repre 
sent  the  distance  between  these 
two  points.  In  the  triangle  formed 
by  the  three  lines  r,  r",  and  D, 
the  angle  V  included  between  r' 
and  r"  will  be  (by  Trigonometry), 
given  by  the  formula, 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  0*7 

^    Msy  =  7-+r"i—l)* 


We  have  only  to  determine  r,  r",  and  D. 

Designating  by  X,  Y,  Z,  the  three  angles  which  the  first 
line  makes  with  the  co-ordinate  axes,  respectively,  and  by 
X',  Y',  Z,  those  made  by  the  second  line,  we  have  by  Art.  76, 

x'  =  r'  cos  X,  \f  =  r'  cos  Y,  z  =  r  cos  Z, 

x"  =  r"  cos  X',          y"  =  r"  cos  Y',          z"  =  r"  cos  Z'. 

Besides,  D  being  the  distance  between  two  points,  we 
have 

D2  =  (x"  —  *')*  +  (y"  -  y'Y  +  (z"  -  z')' 
or 

D2  =  x2  +  y'2  +  z*  +  x"2  +  y"2  +  z"2  —  2  (x' x    +  y' y"  +  z' z"). 

Putting  for  xt  y,  z,  &c.  their  values  in  terms  of  the  angles 
we  have 

D2  =  r'2  |  cos2  X  +  cos2  Y  +  cos2  Zj  +  r"2  Jcos2  X'  +  cos2  Y' 
+  cos2  Z'  I  —  2  r'  r"  _|cos  X  cos  X'  +  cos  Y  cos  Y'  +  cos  Z 
cos  Z'  | . 

But  we  have  (Art.  76), 
cos2  X  +  cos2  Y  +  cos2  Z  =  1,  cos2  X'  +  cos2  Y'  +  cos2  Z'  =  1 ; 

hence 

D2  -  r'2  +  r"2  —  2r  r"  (cos  X  cos  X'  +  cos  Y  cos  Y'  +  cos  Z 

cos  Z'). 

Substituting  this  value  of  D2  in  the  formula  for  the  cosine 
V,  and  dividing  by  2r'  r",  we  have 

cos  V  =  cos  X  cos  X'  -f  cos  Y  cos  Y'  +  cos  Z  cos  Z'; 

which  is  the  expression  for  the  cosine  of  the  angle  formed  in 
space. 


(58  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

87.  We  may  also  express  cos  V  in  functions  of  the  co-effi 
cients  a,  b,  a,  b',  which  enter  into  the  equations  of  the  lines 

x  =  az,         x  ==  a'z, 
y  =  bz,         y  =  b'z. 

For  this  purpose  let  us  consider  the  point  which  we  have 
taken,  on  the  first  line,  whose  co-ordinates  are  x,  y,  zr 
These  co-ordinates  must  have  between  them  the  relations 
expressed  by  the  equations  of  the  line ;  hence 

x  =  az  ^   &   t 

y'  =  bz"; 

and  as  we  have  always  for  the  distance  r' 

'2  '2      i         '2      f        '2  M*^* 

r    =  x    -t-  y    -t-  z  ,  —  4  .     •  ;••/' 

these  three  equations  give 

ar  br'  ,_  r' 


But  we  have 


x'  u'  z' 

cos  X  =  —7-  »     cos  Y  =  -^--  »      cos  Z  =  —  r  ; 
r  r  r 


cos  Z  =  — = 


+a2  +  b2 

Reasoning  in  the   same   manner  on  the  equations  of  the 
second  line,  we  shall  have 

a'  V 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  69 


and  these  values  being  substituted  in  the  general  value  of 
cos  V,  it  becomes 


1  +  aa1  +  bb1 
cos  V  =  ±  — 


/T~4^ — 2    I     12        /  i     , — 73  _|_  7/2" 

This  value  of  cos  V  is  double,  on  account  of  the  double 
sign  of  the  radicals  in  the  denominator.     One  value  belongs 
o  the  acute  angle,  the  other  to  the  obtuse  angle,  which  the 
lines  we  are  considering  make  with  each  other. 

88.  The  different  suppositions  which  we  make  on  the  angle 
V  being  introduced  into  the  general   expression  of  cos  V, 
we  shall  obtain  the  corresponding  analytical  conditions.    Let 
V  =  90°. 

Cos  V  =  o,  and  then  the  equation  which  gives  the  value  of 
cos  V  will  give 

I  +  aa'  +  bb1  =  o, 

which  is  the  condition  necessary  that  the  lines  be  perpendicular 
to  each  other. 

89.  If  the  lines  be  parallel  to  each  other,  cos  V  =  rh  1,  and 
this  gives 

1  +  aa'  -f  bb' 


VI  +az  +  b2    V 1  +  a12  +  b'2 

Making  the  denominator  disappear,  and  squaring  both  mem 
bers,  we  may  put  the  result  under  the  form 

(a  —  a)2  +  (V  —  b)3  +  (aV  —  a'b)2  =  o. 

But  the  sum  of  the  three  squares  cannot  be  equal  to  sero, 
unless  each  is  separately  equal  to  zero,  which  gives 


70  %       ANALYTICAL  GEOMETRY.  [CHAP.  III. 

a  =  a,         b  =  b',         ah'  =  a'b. 

The  two  first  indicate  that  the  projections  of  the  lines  on 
the  planes  of  xz  and  yz  are  parallel  to  each  other;  the  third 
is  a  consequence  of  the  two  others. 

EXAMPLES. 

1.  Find  the  angle    between  the  lines  represented  by  the 
equations 

x~—  2  +  2  x  =  2z  —  3 

and 
y  =  +  z—1  y  =  z  +  2 

Ans.  90°. 

2.  Find  the  angle    between  the  lines  represented  by  the 
equations 

=  Zz  —  3  ^  =  2  z  —  y 


3.  Find  the  angle  between  the  lines  represented  by  the 
equations 

x  =  —2—l  x  =  z  +  2 

r-s  and    y  =  2z-i 

90.  It  is  evident  that  the  angles  X,  Y,  Z,  which  a  straight 
line  makes  with  the  co-ordinate  axes,  are  complements  of  the 
angles  which  the  same  line  makes  with  the  co-ordinate  planes 
respectively  perpendicular  to  the  axes.  Hence,  if  we  desig 
nate  by  U,  Ur,  U",  the  angles  which  this  line  makes  with  the 
planes  of  yz,  xz,  and  xy,  we  shall  have  (Art.  87), 


cos  X  —  sin  U  =  —  =?,  cos  Y  =  sin  U'  = 

1 

cos  Z  =  sin  U"  = 


CHAP.  III.J  ANALYTICAL  GEOMETRY.  71 

91.  Let  it  be  required  to  find  the  conditions  necessary  that 
two  lines  should  intersect  in  space-  and  also  find  the  co-ordi 
nates  of  their  point  of  intersection. 

Let 

x  =  az  +  a,         x  =  a'z  +  a 

y  =  bz  +  (3,         y  =  b'z  +  (3'. 

be  the  equations  of  the  given  lines.  If  they  intersect,  the 
co-ordinates  of  their  point  of  intersection  must  satisfy  the 
equations  of  these  lines  at  the  same  time.  Calling  x',  y',  z't 
the  co-ordinates  of  this  point,  we  have 

x'  =  az'  +  a,         x  =  a'z'  +  a', 

y'  =  bz'  +  /3,         y'  =  b'z  +  13'. 

These  four  equations  being  more  than  sufficient  to  deter 
mine,  the  three  quantities  x',  y',  z',  will  lead  to  an  equation 
of  condition  between  the  constants  a,  b,  a,  /3,  a',  /3',  a',  b', 
which  fix  the  positions  of  the  lines,  which  condition  must 
be  fulfilled  in  order  that  the  lines  intersect.  Eliminating  x 
and  y',  we  have 

(a  —  a')z'  +  a  —  a'  =  o,       (b  —  b')z'  +  @  —  (3'  =  o, 
and  afterwards  z ' ,  \\Q  get 

(a  —  a')  (3  —  j3')  —  (a  —  a')  (b  —  b')  =  o, 

which  is  the  equation  of  condition  that  the  two  lines  should 
intersect.  If  this  condition  be  fulfilled,  we  may,  from  any 
three  of  the  preceding  equations,  find  the  values  of  x',  y',  z, 
and  we  get 

a  —  a  Q'—3  aa  —a'oi  bS'  —  b'j3 

z  = ,    or  z  =  -, TT»    x  = r>    y  =  — ; =-; — • 

a  —  a  b  —  b  a  —  a '       y  b  —  b 

These  values    become    infinite  when   a  =  a'  and   b  =  b'. 


72  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

The  point  of  intersection  is  then  at  an  infinite  distance.     In 
deed,  on  this  supposition  the  lines  are  parallel. 

92.  The  method  which  has  just,  been  applied  to  the  inter 
section  of  two  straight  lines,  may  also  be  used  to  determine 
the  points  of  intersection  of  two  curves  when  their  equations 
are  known.      For  these    points   being  common  to  the  two 
curves,  their  co-ordinates  must  satisfy  at  the  same  time,  the 
equations  of  the  curves.     This  consideration  will  generally 
give  one  more  equation  than  there  are  unknown  quantities. 
Eliminating  the  unknown  quantities,  we  obtain  an  equation 
of  condition  wnich  must  be  satisfied,  in  order  that  the  two 
curves  intersect.    As  the  determination  of  these  intersections 
will  be  better  understood  when  we  have  made  the  discussion 
of  curves,  this  subject  will  be  resumed. 

EXAMPLES. 

1.  Find  the  equations  of  a  straight  line  in  space,  which 
shall  pass  through  a  given  point,  and  be  parallel  to  a  given 
line. 

2.  Find  the  co-ordinates  of  the  points  in  which  a  given 
straight  line  in  space  meets  the  co-ordinate  planes. 

Of  the  Plane. 

93.  We  have  seen  that  a  line  is  characterized  when  we 
have  an  equation  which  expresses  the  relations  between  the 
co-ordinates  of  each  of  its  points.     It  is  the  same  with  sur 
faces,  and  their  character  is  determined  when  we  have  an 
equation  between  the  co-ordinates  x,  y,  and  2,  of  the  points 
which   belong  to  it;    for  by  giving  values  to  two  of  these 
variables,  the  third  can  be  deduced,  which  will  give  a  point 
on  the  surface. 


CHAP.  111.]  ANALYTICAL  GEOMETRY.  73 

94.  The  Equation  of  a  Plane  is  an  equation  which  ex 
presses  the  relations  between  the  co-ordinates  of  every  point 
of  the  plane. 

Let  us  find  this  equation. 

A  plane  may  be  generated  by  considering  it  as  the  locus 
of  all  the  perpendiculars,  drawn  through  one  of  the  points 
of  a  given  straight  line.  Let  x,  y ,  z',  be  the  co-ordinates 
of  this  point,  we  have  (Art.  84), 

x  —  x'  =  a  (z  —  z') 


.  for  the  equations  of  the  given  line. 

y  —  y'  = b  (z  —  O  ) 

Those  of  another  line   drawn    through   the   same   point, 
will  be 

x  —  x'  =  a'  (z  —  z') 

y  —  y'  =  b'(z  —  z'). 

If  these  two  lines  be  perpendicular,  we  have  (Art.  88) 

the  condition 

1  +  aa'  +  bb'  =  o, 

a'  and  b'  being  constants  for  one  perpendicular,  but  variables 
from  one  perpendicular  to  another.  If  we  substitute  for  a 
and  b1  their  values  drawn  from  the  above  equations,  the 
resulting  equation  will  express  a  relation  which  will  corre 
spond  to  all  the  perpendiculars,  and  this  relation  will  be  that 
which  must  exist  between  the  co-ordinates  of  the  plane  which 
contains  them.  The  elimination  gives 

z  —  z'  +  a  (x  —  x')  +  b  (y  —  y')  =  o, 

which  is  the  general  equation  of  a  plane,  since  a  and  b  are 
entirely  arbitrary,  as  well  as  x't  y ',  and  z'. 
95.  If  \ve  make  x  =  o,  and  y  =  o,  we  have 
7  K 


74 


ANALYTICAL  GEOMETRY. 


[CHAP.  Ill 


z  =  z'  +  ax'  +  by' 

for  the  ordinate  of  the 
point  C,  at  which  the 
plane  cuts  the  axis  of  z. 
Representing  this  dis 
tance  by  c,  the  equation 
of  the  plane  becomes 

z  i-  ax  +  by  —  c  =  o, 

and  we  see  that  it  is  linear  with  respect  to  the  variables 
x,  y,  and  z.  It  contains  three  arbitrary  constants,  a,  b,  c, 
because  three  conditions  are,  in  general,  necessary  to  deter 
mine  the  position  of  a  plane  in  space.  If  c  =  o,  the  plane 
passes  through  the  origin. 

96.  To  find  the  intersection  of  this  plane  with  the  plane 
of  xz,  make  y  =  o,  and  we  have 

y  =  o,     z  +  ax  —  c  =  o, 

for  the  equations  of  the  intersection  CD. 

The  first  shows  that  its  projection  on  the  plane  of  xy  is  in 
the  axis  of  x,  and  the  second  gives  the  trigonometrical  tan 
gent  of  the  angle  which  it  makes  with  the  axis  of  x. 

97.  Making  x  =  o,  we   obtain   the   intersection  CD',  the 
equations  of  which  are, 

x  =  o,     z  +  by  —  c  =  o; 
and  z  =  o  gives 

z  =  o,     ax  +  by  —  c  =  o, 

for  the  equations  of  the  intersection  DD'. 

The  intersections  CD,  CD',  DD',  are  called  the  Traces  of 
the  Plane. 


CHAP.  III.  |  ANALYTICAL  GEOMETRY.  75 

98.  The  projections  of  the  line  to  which  this  plane  is  per 
pendicular,  have  for  their  equations 

(x_x>)  =  a(z_z'))      (y  —  y)  =  b  (z  —  %'). 

Comparing  them  with  those  of  the  traces  CD,  CD',  put 
under  the  form 

1  c  I  c 


We  see  (Art.  64)  that  these  lines  are  respectively  perpen 
dicular  to  each  other,  since 

1  +  a  X  ---  =  o, 


and 


1  +  b  X =  o. 

o 


Hence,  if  a  plane  be  perpendicular  to  a  line  in  space,  the 
traces  of  the  plane  will  be  perpendicular  to  the  projections  of 
the  line. 

99.  Making  z  =  o  in  the  equations  of  the  traces  CD,  CD', 

we  have 

c 
z  =  ot    y  =  o,     x  —  —  > 

a 
and 

c 

2  =  0,     a?  =  o,       y=  —9 

for  the  co-ordinates  of  the  points  D,  D',  in  which  the  traces 
meet  the  axes  of  x  and  y.  These  equations  must  satisfy  the 
equations  of  the  third  trace  DD',  because  this  trace  passes 
through  the  points  D  and  D'. 

100.  Let  us  put  the  equation  of  the  plane  under  the  form 

Ax  +  By  +  Cz  +  D  =  o, 
which  is  the  same  form  as  the  preceding,  if  we  divide  by  C. 


76  ANALYTICAL  GEOMETRY.  [CHAP.  IIL 

We  wish  to  show  that  every  equation  of  this  form  is  the 
equation  of  a  plane. 

From  the  nature  of  a  plane,  we  know  that  if  two  points 
be  assumed  at  pleasure  on  its  surface,  and  connected  by  a 
straight  line,  this  line  will  lie  wholly  in  the  plane.  If  we 
can  prove  that  this  property  is  enjoyed  by  the  surface  repre 
sented  by  the  above  equation,  it  will  follow  that  this  surface 
is  a  plane. 

x  =  az  H-  a, 
y  =  bz  +  /3, 

be  the  equations  of  the  line,  and  let  x',  y',  z',  be  the  co-ordi 
nates  of  one  of  the  points  common  to  the  line  and  surface. 
They  must  satisfy  the  equations  of  the  line  as  well  as  that 
of  the  surface,  and  we  have 

a?'  =  az'  +  a,         y'  =  bz'  +  /3, 
and 

Aa?'  +  By'  +  Cz'  +  D  =  o. 

Substituting  for  x'  and  y'  their  values  az'  -f  «,  bz'  +  /3,  we 

have 

(Aa  +  Eb  +  C)  z'  +  Aa  +  B/3  -f  D  =  o, 

which  is  the  equation  of  condition  in  order  that  the  line  and 
surface  have  a  common  point. 

Let  x",  y"  z",  be  the  co-ordinates  of  another  point  common 
to  the  line  and  surface.  We  deduce  the  corresponding  con 
dition 

(Aa  +  Eb  +  C)  z"  -f  Aa  +  B/3  +  D  =o. 

Now,  these  two  equations  cannot  subsist  at  the  same  time, 
unless  we  have  separately 

Aa  +  Eb  +  C  =  o,     and  Aa  +  B/3  +  D  =  o. 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  77 

These  are,  therefore,  the  necessary  conditions  that  the  line 
and  surface  have  two  points  common. 

If  the  values  of  <z,  b,  a,  /3,  are  such  that  these  two  condi 
tions  are  satisfied,  every  point  of  the  line  will  be  common  to 
the  surface.  For,  if  x"',  y'",  z'",  be  the  co-ordinates  of  an 
other  point,  in  order  that  it  be  on  the  surface,  we  must  have 

(A<7  +  B6  +  C)  z"'  +  AOL  +  ES  +  D  =  o. 

But  this  equation  is  satisfied  whenever  the  two  others  are, 
and  consequently  this  point  is  also  common  to  the  line  and 
surface. 

As  the  same  may  be  proved  for  every  other  point,  it  fol 
lows  that  every  straight  line  which  has  two  points  in  common 
with  the  surface  whose  equation  is 

Az  +  Ey  +  Cz  +  D  =  o, 

will  coincide  with  it,  and  consequently  this  surface  is  a  plane. 
101.  If  we  make  y  —  o,  we  have 

Ax  +  Cz  +  D  =  o 

for  the  equation  of  the  trace  CD,  on  the  plane  xz.  If  the 
plane  be  perpendicular  to  the  plane  of  yz,  this  trace  will  be 
parallel  to  the  axis  of  x,  and  its  equation  wrill  be  of  the  form 
z  =  a,  which  requires  that  A  =  o,  and  the  equation  of  the 

plane  becomes 

By  +  Cz  +  D  =  o. 

We  should  in  like  manner  have  B  =  o,  if  the  plane  were 
perpendicular  to  the  plane  of  xz.  Its  trace  on  the  plane  of 
t/z  would  be  parallel  to  the  axis  of  y,  and  its  equation 
would  be 

Ax  +  Cz  f  D  =  o. 

For  a  plane  perpendicular  to  the  plane  of  xy,  we  have  the 

equation 

7* 


78  ANALYTICAL  GEOMETRY.  [CHAP.  Ill 

Ax  +  Ey  +  D  =  o, 

This  condition  requires  that  we  have  C  —  o. 

We  may  readily  see  that  these  different  forms  result  from 

the  fact  that  —  -^  »  —  ^  '    represent    the    trigonometrical 

tangents  of  the  angles  which  the  traces  on  the  planes  of  xz 
and  yz  make  with  the  axes  of  x  and  y. 

102  There  are  many  problems  in  relation  to  the  plane 
which  may  be  resolved  without  difficulty  after  what  has 
been  said.  We  will  examine  one  or  two  of  them. 

Let  it,  be  required  to  find  the  equation  of  a  plane  passing 
through  three  given  points. 

Let  x',  y',  z' ;   x",  y",  z" ;  x'",  y'",  z" ;    be  the  co-ordinates 

of  these  points, 

Ax  +  Ey  +  Cz  +  D  =  o, 

will  be  the  form  of  the  equation  of  the  required  plane. 
Since  this  plane  must  pass  through  the  three  points,  we  will 
have  the  relations 

.     A*'  +  By'  +  Cz'   +  D  =  o, 
Ax"  +  Ey"  +  Cz"  +  D  =  o, 
Ax"'  +  Ey'"  +  Cz"  +  D  =  o. 
Then  these  equations  will  give  for  A,  B,  C,  expressions  of 

the  form 

A  =  A'D,    B  =  B'D,    C-C'D, 

A',  B',  C',  being  functions  of  the  co-ordinates  of  the  given 
points. 

Substituting  these  values  in  the  equation  of  the  plane,  we 

have 

A'x  +  E'y  +  C'z  +  1  =  o, 

for  the  equation  of  a  plane  passing  through  three  given 
points. 


CHAP.  111.]  ANALYTICAL  GEOMETRY.  79 

103.  To  find  the  intersection  of  two  planes  represented 
by  the  equations 

Ax  +  By  +  Cz  +  D  =  o, 
A'x  +  B'y  +  C'z  +  D'  =  o. 

These  equations  must  subsist  at  the  same  time  for  the 
points  which  are  common  to  the  two  planes.  We  may  then 
determine  these  points  by  combining  these  equations. 

If  we  eliminate  one  of  the  variables,  z  for  example,  we 
have 

(AC'  —  A'C)  x  +  (EC'  —  B'C)  y  +  (DC'  —  D'C)  =  o. 

This  equation  being  of  the  first  degree,  belongs  to  a 
straight  line.  It  represents  the  equation  of  the  projection 
of  this  intersection  on  the  plane  of  xy. 

By  eliminating  x  or  y,  we  can  in  a  similar  manner  find 
the  equation  of  its  projection  on  the  planes  of  yz  and  xz. 

104.  Generalizing  this  result,  we  may  find  the  intersections 
of  any  surfaces  whatever.      For,  as    their  equations    must 
subsist  at  the  same  time  for  the  points  which  are  common, 
by  eliminating  either  of  the  variables,  the  resulting  equations 
will  be  those  of  the  projections  of  the  intersections  on  the 
co-ordinate  planes. 

Of  the   Transformation  of  Co-ordinates. 

105.  We  have  seen  that  the  form  and  position  of  a  curve 
are  always  expressed  by  the  analytical  relations  which  exist 
between  the  co-ordinates  of  its  different  points.     From  this 
fact,  curves  have  been  classified  into  different  orders  from 
the  degree  of  their  equations. 


80  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

106.  Curves  are  divided  into  algebraic  and  transcendental 
urves. 

Algebraic  Curves  are  those  whose  equations  are  purely 
algebraic. 

Transcendental  Curves  are  those  whose  equations  are  ex 
pressed  in  terms  of  logarithmic,  trigonometrical,  or  expo 
nential  functions. 

if  =  a2  —  x*  is  an  algebraic  curve. 

y  =  sin  x,  y  —  cos  x,  y  =  ax,  &c.,  are  transcendental 
curves. 

107.  Algebraic  Curves  are   classified    from  the  degree  of 
their  equation,  and  the  order  of  the  curve  is  indicated  by  the 
exponent  of  this  degree.     For  example,  the  straight  line  is 
of  the  first  order,  because  its  equation  is  of  the  first  degree 
with  respect  to  the  variables  x  and  y. 

108.  The  discussion  of  a  curve  consists  in  classifying  it 
and  determining   its    position  and  form  from  its  equations, 
This  discussion  may  be  very  much  facilitated  by  means  of 
analytical  transformations,  which,  by  simplifying  the  equa 
tions  of  the   curve,  enable  us  more  readily  to  discover  its 
form  and  general    properties.      The  methods  used  to  effect 
this    simplification  consist  in  changing   the    position  of  the 
origin,  and  the  direction  of  the  co-ordinate  axes,  so  that  the 
proposed   equations,  when  referred  to  them,  may  have  the 
simplest  form  of  which  the  nature  of  the  curve  will  admit. 

109.  When  we  wish  to  pass  from  one  system  of  co-ordi 
nates  to  another,  we  find,  for  any  point,  the  values  of  the  old 
co-ordinates  in  terms  of  the  new.     Substituting  these  values 
in  the  proposed   equation,  it  will  express  the  relations  be 
tween  the  co-ordinates  of  the  same  points  referred  to  this 


CHAP.  HI.]  ANALYTICAL  GEOMETRY.  81 

new  system.  Consequently  the  properties  of  the  curve  will 
remain  the  same,  as  we  have  only  changed  the  manner  of 
expressing  them. 

110.  The  relations  between  the  new  and  old  co-ordinates 
are  easily  established,  when 

the  origin  alone  is  changed 
without  altering  the  direc 
tion  of  the  axes.  For,  let 
A'  be  the  new  origin,  and 
A'X',  A'Y,  the  new  axes, 
parallel  to  the  old  axes,  AX, 
AY'.  For  any  point  M,  we 
have 

AP  =  AB  +  BP,    PM  =  PP'  +  P'M  =  A'B  +  P'M. 
Making  AB  =  a,  and  A'B  =  6,  and  representing  by  x  and 
y  the  old,  and  x',  y'  the    new  co-ordinates,  these  equations 

become 

x  =  a  +  x',    y  =  b  +  y'9 

wnich  are  the  equations  of  transformation  from  one  system 
of  co-ordinate  axes,  to  another  system  parallel  to  the  first. 

111.  To  pass  from  one  system  of  rectangular  co-ordinates 
to  another  system  oblique  to  the  first,  the  origin  remaining 
the  same. 

Let  AY,  AX,  be  two  axes  at 
right  angles  to  each  other,  and 
AY',  AX',  two  axes  making  any 
angle  with  each  other.  Through 
any  point  M,  draw  MP,  MP', 
respectively  parallel  to  AY  and 
AY',  and  through  P  draw  P'Q,  P'R  parallel  to  AX  and  AY, 
we  shall  have 


82  ANALYTICAL  GEOMETRY.  [CHAP.  IIL 

a?  =  AP  =  AR  +  P'Q,  '  y  =  MP  =  MQ  +  P'R. 

But  AR,  P'R,  MQ,  PQ,  are  the  sides  of  the  right-angled 
triangles  APR,  P'MQ,  in  which  AP'  =  x,  and  P'M  =  yf. 
We  also  know  the  angles  P'AR  =  a  and  MP'Q  =  a'.  We 
deduce  from  these  triangles 

x  =  x  cos  a  +  y  cos  a',         y  =  xf  sin  a  +  y  sin  a', 

which  are  the  relations  which  subsist  between  the  co-ordi 
nates  of  the  two  systems. 

112.  If  we  wished  to  pass  from  the  system  whose  co-ordi 
nates  are  x'  and  y  to  that  of  x  and  y,  we  have  only  to  de 
duce  the  values  x  and  y'  from  the  two  last  equations.  We 
find  by  elimination  these  values  to  be 

x  sin  a'  —  11  cos  a'  v  cos  a  —  x  sin  a 

,/     -  -  J  -     J 


.  __ 

sin  (a  —  a)  sin  (a  —  a) 

If  the  new  axes  of  x'  and  ?/'  be  rectangular  also,  we  have 
a  —  a  =  90°  and  a'  =  90°  +  a,    sin  (a  —  a)  =  sin  90°  =  1. 
sin  a'  =  sin  (90°  +  a)  =  sin  a  cos  90°  -f  cos  a  sin  90°  =  cos  a, 
cos  a'  =  cos  90°  cos  a  —  sin  90°  sin  a  =  —  sin  a. 

Substituting  these  values,  we  have  for  the  formulas  for 
passing  from  a  system  of  rectangular  co-ordinates  to  another 
system  also  rectangular,  the  origin  remaining  the  same, 

x  =  x  cos  a  —  y'  sin  a,     y  =  x'  sin  a  +  y'  cos  a. 

113.  To  pass  from  a  system  of  oblique  co-ordinates  to 
another  system  also  oblique,  the  origin  remaining  the  same. 

Let  AX',  AY'  be  the  axes  of  x',  y,  and  AX",  AY",  the  new 
axes  whose  co-ordinates  are  x",  y".  Let  us  take  a  third 
system  at  right  angles  to  each  other  as  AX,  AY,  the  co-or- 


CHAP.  III.] 


AX ALYTICAL  GEOMETRY. 


dinates    being  x,  y.     Calling  a,  a', 
\n  i*  ^ 

8,  p',  the  angles  which  the  axes  of 
a:',  y,  x",  y",  make  with  the  axis  of 
x,  we  have  (Art.  Ill)  for  passing 
from  this  system  to  the  two  systems 
of  oblique  co-ordinates,  the  formulas 

x  =  x'  cos  a  +  y  cos  a',         y  =  x  sm  a  +  y'  sin  a', 
x  =  x"  cos  J3  +  y'  cos  /3',         ?/  =  a?"  sin  ,3  +  y"  sin  /3'. 

Eliminating  a:  and  y  from  these  equations,  we  shall  obtain 
the  equations  which  will  express  the  relations  between  the 
co-ordinates  x't  y',  and  x",  y",  which  are 

x'  cos  a  +  y'  cos  a'  =  a;"  cos  /3  +  y"  cos  j3' 
x'  sin  a  +  y'  sin  a'  =  a?"  sin  /3  +  y"  sin  /3'. 

Multiplying  the  first  by  sin  a,  and  subtracting  from  it  the 
second  multiplied  by  cos  a,  we  obtain  the  value  of  y'. 
Operating  in  the  same  manner,  we  get  the  value  of  x't  and 
the  formulas  become 

,  _  x"  sin  (a'  —  /3)  -f  y"  sin  (af  —  /3f) 

X    —  ;  -  —  -  —  -  -  r  -  j 

sin  (a  —  a) 

,  _  x"  sin  (g  —  a)  +  y"  sin  (.3'  —  a) 
sin  (a  —  a) 

114.  Generalizing  the  foregoing  remarks,  we  may  easily 
find  the  formulas  for  the  transformation  of  co-ordinates  in 
space.  We  have  only  to  find  the  value  of  the  old  co-ordi 
nates  in  terms  of  the  new,  and  reciprocally.  If  the  trans 
formation  be  to  a  parallel  system,  and  a,  b,  c,  represent  the 
co-ordinates  of  the  new  origin,  we  have  the  formulas 

x  =  a  +  x',    y  =  b  +  y',     z  =  c  +  z', 


84  ANALYTICAL  GEOMETRY,  [CHAP.  III. 

in  which  x,  y,  and  z,  are  the  old,  and  x',  y ',  and  z',  the  new 
co-ordinates. 

115.  Let  us  now  suppose  that  the  direction  of  the  new 
axes  is  changed.  As  the  introduction  of  the  three  dimen 
sions  of  space  necessarily  complicates  the  constructions  of 
the  problems,  if  we  can  ascertain  the  form  of  the  relations 
which  must  exist  between  the  old  and  new  co-ordinates,  this 
difficulty  may  be  obviated. 

Now  it  can  be  proved,  in  general,  that  in  passing  from  any 
system  of  co-ordinates,  the  old  co-ordinates  must  always  be 
expressed  in  linear  functions  of  the  new,  and  reciprocally. 
This  has  been  verified  in  the  system  of  co-ordinates  for  a 
plane,  since  the  relations  which  we  have  obtained  are  of  the 
first  degree.  To  show  that  this  must  also  be  the  case  with 
transformations  in  space,  let  us  conceive  the  values  of  #,  y,  z, 
expressed  in  any  functions  of  a?',  y,  z',  which  we  will  designate 
by  <p,  if,  -^,  so  that  we  have 

x  =  <p  (x',  y',  z'),  y  =  «  (x',  y',  z'),  z  =  ^  (x1 ,  ?/',  z'). 

If  we  substitute  these  values  in  the  equation  of  the  plane, 
which  is  always  of  the  form 

Ax  +  By  +  Cz  +  D  =-  o, 
it  becomes 

A.  <p  (a?',  y',  z',)  +  B.  «•  (x',  y,  z',)  +  C.  ±  (x,  y,  z',)  +  D  =  o. 

But  the  equation  of  the  plane  is  always  of  the  first  degree, 
whatever  be  the  direction  of  the  rectilinear  axes  to  which  it 
is  referred,  since  the  equations  of  its  linear  generatrices  are 
always  of  the  first  degree.  Hence,  the  preceding  equations 
must  reduce  to  the  form 

AV+  By  +  C'z  +  D'  =  o, 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  85 

in  which  A',  B',  C'  D',  are  independent  of  x',  \j ',  z',  but  de 
pendent  upon  the  primitive  constants  A,  B,  C,  D,  and  the 
angles  and  distances  which  determine  the  relative  positions 
yf  the  two  systems. 

This  reduction  must  take  place  whatever  be  the  values  of 
the  primitive  co-efficients  A,  B,  C,  D,  and  without  there  re 
sulting  any  condition  from  them.  Hence  this  reduction 
must  exist  in  the  functions  9,  *,  4,,  themselves,  for  if  it  were 
otherwise,  the  terms  of  <p  which  are  multiplied  by  .A,  would 
not,  in  general,  cause  those  of  t  and  ^  to  disappear,  which 
are  multiplied  by  B  and  C.  It  would  follow  from  this,  that 
the  powers  of  x',  y',  z,  higher  than  the  first,  would  necessa 
rily  remain  in  the  transformed  equation,  if  they  existed  in 
the  functions  <p,  cr,  4,.  These  functions  are  therefore  limited 
by  the  condition  that  the  new  co-ordinates  x,  y,  z,  exist 
only  of  the  first  power,  and  consequently  the  most  general 
form  which  we  can  suppose,  will  be 

x  =  a  +  mx'  +  m'y'  +  m"z', 

y  =  b  +  nx'  +  riy1  +  ri'z', 
z  =  c  +  px'  +  p'y'  +  p"z'9 

in  which  the  co-efficients  of  x',  y',  z,  are  unknown  constants 
which  it  is  required  to  determine.  But  since  they  are  con 
stants,  their  values  will  remain  always  the  same,  whatever 
be  those  of  x',  y',  z'.  We  can  then  give  particular  values  to 
these  variables,  and  thus  determine  those  of  the  constants. 
If  we  make 

x'  =  o,    y'  =  o,     z'  =  o, 
we  have 

x  =  a,     y  =  b,     z  =  c, 

which  are  the  co-ordinates  of  the  new  origin  with  respect  to 
8 


86 


ANALYTICAL  GEOMETRY. 


[CHAP.  Ill 

the  old.  We  will  suppose  for  more  simplicity  that  the  di 
rection  of  the  axes  is  changed,  without  removing  the  origin; 
the  preceding  formulas  become  under  this  supposition 

x  =  mx'  +  m'y'  +  m"z', 
y  =  nx'  +  riy'  4-  n"z'9 
z  =  px'  4-  p'y  4-  p"z'. 

To  determine  the  constants,  let  us  consider  the  points 
placed  on  the  axis  of  x'9  the  equations  of  this  axis  are 

y'  =  o,         z1  =  o, 
We  have  then  for  points  situated  on  it, 

x  =  mx'    y  =  nx',     z  =  px'. 

Let  AX'  be  this  axis, 
and  let  the  old  axes  AX, 
AY,  AZ,  be  taken  at 
right  angles,  for  any  point 
M  we  have  AM  =  x't 
MM'  =  z,  and  the  triangle 
AMM'  will  give 

z  =  x'  cos  AMM', 

The  angle  AMM'  is  that  which  the  new  axis  of  x'  makes 
with  the  old  axis  of  z.  Let  us  call  it  Z,  and  represent  by 
X  and  Y,  the  angles  formed  by  this  same  axis  AX',  with  AX 
and  AY.  We  shall  have  for  points  on  this  axis, 

x  —  x  cos  X,     y  =  x'  cos  Y,     z  —  x'  cos  Z. 
This  result  determines  n,  m,  p,  and  gives 

m  =  cos  X,     n  =  cos  Y,    p  —  cos  Z. 
If  we  considei  points  on  the  axis  of?/',  whose  equations  are 
x'  —  o,          z'  =  o,  . 


CHAP.  III.]  ANALYTICAL  GEOMETRY.  87 

we  shall  have  relatively  to  these  points 

x  =  my',       y  =  riy,       z  =  p'y. 

Designating  by  X',  Y',  Z',  the  angles  which  this  axis  forms 
with  the  axis  of  x,  y,  z,  we  have 

m'  =  cos  X',     ri  =  cos  Y',    p'  =  cos  Z'. 
Reasoning  in  the  same  manner  with  the  axis  z',  we  have 

m"  =  cos  X",     n"  =  cos  Y",    p"  =  cos  Z" ; 
from  which  we  get 

x  =  x  cos  X  +  y  cos  X'  +  z  cos  X'7, 
y  =  x  cos  Y  +  y  cos  Y'  +  z'  cos  Y", 
z  =  x  cos  Z  +  y  cos  Z'  +  z  cos  Z".  (1) 

116.  We  must  join  to  these  values,  the  equations  of  con 
dition  which  take  place  between  the  three  angles,  which  a 
straight  line   makes  with   the   three   axes,  and  which   are 
(Art.  76),  UO 

cos2  X  +  cos2  Y  +  cos2  Z  =  1, 

cos2X'  +  cos2Y'  +  cos2Z'  =  1, 

cos2  X"  +  cos2  Y"  +  cos2  Z"  =  1.  (2) 

These  formulas  are  sufficient  for  the  transformation  of  co 
ordinates,  whatever  be  the  angles  which  the  new  axes  make 
with  each  other. 

117.  Should  it  be  required  that  the  new  axes  make  par 
ticular  angles  with  each  other,  there  will  result  new  condi 
tions  between  X,  Y,  Z,  X',  &c.,  which  must  be  joined  to  the 
preceding   equations.      If   we   represent    by   V   the    angle 
formed  by  the  axis  of  x  with  that  of  y,  by  U  that  made  by 
if  with  z,  and  by  W  that  made  by  z  with  x',  we  have  by 
-\rt.86 


88  ANALYTICAL  GEOMETRY.  [CHAP.  III. 

cos  V  =  cos  X  cos  X'  +  cos  Y  cos  Y'  +  cos  Z  cos  Z', 
cos  U  =  cos  X'  cos  X"  +  cos  Y'  cos  Y"  +  cos  Z'  cos  Z". 
cos  W  =  cos  X  cos  X"  +  cos  Y  cos  Y"  +  cos  Z  cos  Z",  (3) 

And  these  equations  added  to  those  of  (1)  and  (2),  will  enable 
us  in  every  case  to  establish  the  conditions  relative  to  the 
new  axes,  in  supposing  the  old  rectangular. 

118.  If,  for  example,  we  wish  the  new  system  to  be  also 
rectangular,  we  shall  have 

cos  V  =  o,         cos  U  =  o9         cos  W  =  o, 

and  the  second  members  of  equations  (3)  will  reduce  to  zero; 
then  adding  together  the  squares  of  x,  y,  z,  we  find 

T*  +  ff  -f  %2  =  x'*  +  y'2  +  z2. 

This  condition  must  in  fact  be  fulfilled,  for  in  both  sys 
tems  the  sum  of  the  squares  of  the  co-ordinates  represents 
the  distance  of  the  point  we  are  considering,  from  the  com 
mon  origin. 

119.  If  we  wished  to  change  the  direction  of  two  of  the 
axes  only,  as,  for  example,  those  of  x  and  y,  let  us  suppose 
that  they  make  an  angle  V  with  each  other,  and  continue 
perpendicular  to  the  axis  of  z.     We  have  from  these  con 
ditions, 

cos  U  =  o,         cos  W  =  o, 

cos  X"  =  o,        cos  Y"  =  o,         cos  Z"  =  1. 
Substituting  these  values  in  equations  (3),  we  have 
cos  Z'  =  o,         cos  Z  =  o, 

that  is,  the  axes  of  x  and  y  are  in  the  plane  of  xy 
From  this  and  equations  (2),  there  results 

cos  Y  =  sin  X,       cos  Y  =  sin  X', 


AP.  in.]  ANALYTICAL  GEOMETRY.  89 

and  the  values  of  x,  and  y,  become 

x  =  x  cos  X  -f  y'  cos  X',     y  =  x'  sin  X  +  y'  sin  X' ; 
which  are  the  same  formulas  as  those  obtained  (Art.  !!!)• 

Polar  Co-ordinates. 

120.  Right  lines  are  not  the  only  co-ordinates  which  may 
be  used  to  define  the  position  of  points  in  space.    We  may 
employ  any  system  of  lines,  either  straight  or  curved,  whose 
construction  will  determine  these  points. 

For  example,  we  may  take  for  the  co-ordinates  of  points 
situated  in  a  plane,  the  distance  AM, 
from  a  fixed  point  A  taken  in  a  plane, 
and  the  angle  MAX,  made  by  the 
line  AM  with  any  line  AX  drawn  in 
the  same  plane.  For,  if  we  have  the 

Af  ~P  3*' 

angle  MAP,  the  direction  of  the  line 

AM  is  known ;  and  if  the  distance  AM  be  also  known,  the 

position  of  the  point  M  is  determined. 

121.  The    method  of  determining   points  by  means  of  a 
variable  angle   and   distance,  is   called  a   System   of  Polar 
Co-ordinates.     The  distance  AM  is  called  the  Radius  Vector, 
and  the  fixed  point  A  the  Pole. 

122.  When  we  know  the  equation  of  a  line,  referred  to 
rectilinear  co-ordinates,  we  may  transpose  it  into  polar  co 
ordinates,  by  determining  the  values  of  the  old  co-ordinates 
in  terms  of  the  new,  and  substituting  them  in  the  proposed 
equation.     For  example,  let  A'  be  taken 

as  the  pole,  whose  co-ordinates  are  x  =  a, 
y  —  b.  Draw  A'X'  parallel  to  the  axis 
of  x,  and  designate  the  angle  MA'X'  by 
v,  the  radius  vector  A'M  by  r,  we  have 
8*.  M 


00  ANALYTICAL  GEOMETRY.  [CHAP.  1IL 

AX-AB  +  A'Q,    PM  =  A'B  +  MQ, 
or, 

x  =  a  +  A'Q,        y  =  b  +  MQ. 

Bat  in  the  right-angled  triangle  A'MQ,  we  have 

A'Q  =  r  cos  v,  and  MQ  =  r  sin  v. 
Substituting  these  values,  we  have 

x  —  a  -f  r  cos  v,    y  =  b  -f  r  sin  v,     (1) 

which  are  the  formulas  for  passing  from  rectangular  co-ordi 
nates  to  polar  co-ordinates. 

123.  If  the  pole   coincide  with  the  origin,  a  =  o,  b  =  ot 
and  we  have 

x  =  r  cos  v,     y  =  r  sin  v. 

If  the  line  AX'  make  an  angle  a  with  the  axis  of  x, 
formulas  (1)  will  become 

x  =  a  +  r  cos  (v  +  a),     y  =  b  +  r  sin  (v  +  a). 

124.  By  giving  to  the  angle  v  every  value  from  o  to  360°, 
and  varying  the  radius  vector  from  zero  to  infinity,  we  may 
determine  the  position  of  every  point  in  a  plane.     But  from 
the  equation 

x  =  r  cos  v 
we  get 

x 
r  = • 

COS  V 

Now,  since  the  algebraic  signs  of  the  abscissa  and  cosine 
vary  together,  that  is,  are  both  positive  in  the  first  and  fourth 
quadrants,  and  negative  in  the  second  and  third,  it  follows 
that  the  radius  vector  can  never  be  negative,  and  we  conclude 
that  should  a  problem  lead  to  negative  values  for  the  radius 
vector,  it  is  impossible. 


CHAP.  Ill] 


ANALYTICAL  GEOMETRY. 


91 


125.  Polar  co-ordinates  may  also  be  used  to  determine  the 
position  of  points  in  space.  For  this  purpose  we  make  use 
of  the  angle  which 
the  radius  vector  A3I 
makes  with  its  pro 
jection  on  the  plane 
of  xy,  for  example, 
and  that  which  this 
projection  makes  with 
the  axis  of  a:.  MAM' 
is  the  first  of  these 
angles,  and  M'AP  the 
second.  Calling  them 
9  and  d,  and  repre 

senting  the  radius  vector  AM  by  r,  and  its  projection  AM' 
by  r,  we  have 

AP  =  AM'  cos  M'AP, 

x  =  r  cos  &  ; 

=  AM'sinM'AP, 

y  =  r  sin  6  ; 
MM'  =  AM  sin  MAM', 

z  =  r  sin  9. 
We  have  also 

AM'  =  AM  cos  MAM', 

r'  =  r  cos  9, 

from  which  equations  we  deduce 


or 


or 


or 


formulae  which  may  be  applied  to  every  point,  by  attributing 
to  the  variables  6,  9,  and  r,  every  possible  value. 


ANALYTICAL  GEOMETRY. 


[CHAP.  IV 


CHAPTER  IV. 


OF  THE  .CONIC  SECTIONS. 

126.  IP  a  right  cone  with  a  circular  base,  be  intersected 
by  planes  having  different  positions  with  respect  to  its  axis, 
the  curves  of  intersection  are  called  Conic  Sections.  As  this 
common  mode  of  generation  establishes  remarkable  analo 
gies  between  these  curves,  we  shall  employ  it  to  find  their 
general  equation. 

Let  O  be  the  origin  of  a  system  of  rectangular  co 
ordinates  OX,  OY,  OZ.  If 
the  line  AC  at  the  distance 
OC  =  C  from  the  origin,  re 
volve  about  the  axis  OZ, 
making  a  constant  angle  v 
with  the  plane  of  xy,  it  will 
generate  the  surface  of  a 
right  cone  with  a  circular 
base,  of  which  C  will  be  the 
vertex  and  CO  the  axis.  The 
part  CA  will  generate  the 
lower  nappe,  CA'  the  upper 
nappe  of  the  cone.  To  find 
the  equation  of  this  surface. 

The  equation  of  a  line  passing  through  the  point  C,  r,vh<?3Q 
co-ordinates  are 

x'  —  o,    y'  —  o,     z'  =  £, 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  93 

Ltt 

is  of  the  form  (Art.  84), 

x  =  a  (z  —  c),     y  —  b  (z  —  c) ; 

the  co-efficients  a  and  b  being  constants  for  the  same  position 
of  the  generatrix,  but  variables  from  one  position  to  another 
But  we  have  (Art.  90),  }> 

Sin2u  =  ______ 

from  which  we  obtain 

(a2  +  b2)  tang  2v  =  1. 

Substituting  for  a  and  b,  their  values  drawn  from  the  equa 
tion  of  the  generatrix,  we  shall  have 

(1f  +  x2)lzngv  =  (z  —  c)2. 

This  equation  being  independent  of  a  and  b,  it  corresponds 
to  every  position  of  the  line  AC  in  the  generation :  it  is  there 
fore  the  equation  of  the  conic  surface. 

127.  Let  this  surface  be  intersected  by  a  plane  BOY, 
drawn  through  the  origin  O,  and  perpendicular  to  the  plane 
of  xz.  Designating  by  u  the  angle  BOX  which  it  makes 
with  the  plane  of  xy,  its  equation  will  be  the  same  as  that  of 
its  trace  BO,  that  is 

%  =  x  tang  u. 

If  we  combine  this  equation  with  that  of  the  conic  surface, 
we  shall  obtain  the  equations  of  the  projections  of  the  curve 
of  intersection  on  the  co-ordinate  planes.  But  as  the  pro 
perties  of  the  curve  may  be  better  discovered,  by  referring 
it  to  axes,  taken  in  its  own  plane,  let  us  find  its  equation  re 
ferred  to  the  two  axes  OB,  OY,  which  are  situated  in  its 
plane,  and  at  right  angles  to  each  other.  Calling  x'  y'  the 
co-ordinates  of  any  point,  the  old  co-ordinates  of  which 


94  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

were  x,  y,  z,  we  shall    have    in    the    right-angled    triangle 
OPP', 

x  =  OP  =  x  cos  u,     z  —  PP'  =  x'  sin  u  ; 

and  since  the  axes  of  y  and  y'  coincide,  we  shall  also  have 


Substituting  these  values  for  x,  y,  z,  in  the  equation  of  the 
surface  of  the  cone,  we  shall  obtain  for  the  equation  of  inter 
section, 

y'2  tang  2v  +  x'2  cos  zu  (tang  2v  —  tang  2ii)  +  %cx'  sin  u  —  c2; 
or  suppressing  the  accents, 

y2  tang  2v  +  x2  cos  2u  (tang2?;  —  tang  2u)  +  %cx  sin  u  =  c2. 

128.  In  order  to  obtain  the  different  forms  of  the  curves 
of  intersection  of  the  plane  and  cone,  it  is  evident  that  all 
the  varieties  will  be  obtained  by  varying  the  angle  u  from  o 
to  90°.  Commencing  then  by  making 

tt  =  0, 

which  causes  the  cutting  plane  to  coincide  with  the  plane  of 
xy,  the  equation  of  the  intersection  becomes 


which  shows  that  all  of  its  points  are  equally  distant  from 
the  axis  of  the  cone.  The  intersection  therefore  is  a  circle, 
described  about  O  as  a  centre  and  with  a  radius  equal 

c 
}  tang  v 

129.  Let  u  increase,  the  plane  will  intersect  the  cone  in 
a  re-entrant  curve,  so  long  as  u  <^  v,  which  will  be  found 


HAP.  IV.] 


ANALYTICAL  GEOMETRY. 


95 


entirely  on  one  nappe  of  the  cone.  But  u  <^  v  makes  tang 
u  <^  tang  v,  and  the  co-efficients  of  a;2  and  if  will  be  positive 
in  the  equation  of  intersection.  This  condition  characterizes 
a  class  of  curves,  called  Ellipses. 

130.  When   u  =  v,  the  cutting   plane  is  parallel  to  CD. 
The  curve  of  intersection  is  found  limited  to  one  nappe  of 
the  cone,   but   extends   indefinitely  from  B  on  this  nappe. 
The   condition  u  =  v  causes 
the  co-efficient  of  a?2  to  dis 
appear,  and  the  general  equa 
tion  of  intersection   reduces 
to 

y2  tan  2v  +  2cr  sin  u  =  c2. 

These    curves    are    called 
Parabolas. 


131.  Finally,  when  u  >  v,  the 
cutting  plane  intersects  both  nappes 
of  the  cone,  and  the  curve  of  inter 
section  will  be  composed  of  two 
branches,  extending  indefinitely  on 
each  nappe.  In  this  case  tang  u  > 
tang  v,  and  the  co-efficient  of  3? 
becomes  negative.  This  condition 
characterizes  a  class  of  curves  called 
Hyperbolas. 


132.  If  we  suppose  the  cutting  plane  to  pass  through  the 
vertex  of  the  cone,  the  circle  and  ellipse  will  reduce  to  a 
point,  the  parabola  to  a  straight  line,  and  the  hyperbola  to 


96  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

two  straight  lines  intersecting  at  C.  This  becomes  evident 
from  the  equations  of  these  different  curves,  by  making 
c  =  o,  and  also  introducing  the  condition  of  u  being  less 
than,  equal  to,  or  greater  than,  v. 

We  will  now  discuss  each  of  these  classes  of  curves,  and 
deduce  from  their  general  equation  the  form  and  character 
of  each  variety. 

Of  the  Circle. 

133.  If  a  right  cone  with  a  circular  base  be  intersected 
by  a  plane  at  a  distance  c,  from  the  vertex,  and  perpendicular 
to  the  axis,  we  have  found  for  the  equation  of  intersection 
(Art.  128), 

*f+X2  =  7-C2—  ' 

tangzi> 

c2 

Representing  the  second  member 5-  by  R2,  we  have 

tang  v     J 

x*  +  y*  =  R2. 

In  this  equation,  the  co-ordinates  x  and  y  are  rectangular, 
the  quantity  \/x2  +  y2  expresses  therefore  the  distance  of 
any  point  of  the  curve  from  the  origin  of  co-ordinates 
(Art.  59).  The  above  equation  shows  that  this  distance  is 
constant.  The  curve  which  it  represents  is  evidently  the 
circumference  of  a  circle,  whose  centre  is  at  the  origin  of 
co-ordinates,  and  whose  radius  is  R. 

134.  To  find  the  points  in  which  the  curve  cuts  the  axis 
of  oc,  make  y  =  o,  and  we  have 


which  shows  that  it  cuts  this  axis  in  two  different  points, 


C.1AP.   IV.] 


ANALYTICAL  GEOMETRY. 


97 


one  on  each  side  of  the  origin,  and  at  a  distance  R  from  the 
axis  of  y.  Making  x  =  o,  we  find  the  points  in  which  it  cuts 
the  axis  of  y.  We  get 

which  shows  that  the  curve  cuts  this  axis  in  two  points,  one 
above  and  the  other  below  the  axis  of  x,  and  at  the  same 
distance  R  from  it. 

135.  To  follow  the  course  of  the  curve  in  the  intermediate 
•points,  find  the  value  of  y  from  its  equation,  we  get 


These  values  being  equal  and  with  contrary  signs,  it 
follows  that  the  curve  is  symmetrical  with  respect  to  the 
axis  of  x.  If  we  suppose  x  positive  or  negative,  the  values 
of  y  will  increase  as  those  of  x  diminish,  and  when  x  =  o 
we  have  y  =  ±  R,  which  gives  the  points  D  and  D'.  As  x 
increases,  y  will  diminish,  and  when 
x  =  zh  R  the  values  of  y  become  zero. 
This  gives  the  points  B  and  B'.  If 
x  be  taken  greater  than  R,  y  be 
comes  imaginary.  The  curve  therefore 
does  not  extend  beyond  the  value  of 
*==hR. 

136.  The  equation  of  the  circle  may  be  put  under  the  form, 

jf  =(R  +  *)(R  —  x). 

R  -r  x,  and  R  —  x,  are  the  segments  B'P  and  BP,  into  which 
the  ordinate  y  divides  the  diameter.  This  ordinate  is  there 
fore  a  mean  proportional  between  these  two  segments. 

137.  Two  straight  lines  drawn  from  a  point  on  a  curve  to 
9  N 


98 


ANALYTICAL  GEOMETRY. 


[CHAP.  IV 

the  extremities  of  a  diameter,  are  called  supplemental  chords. 
The  equation  of  a  line  passing  through 
the  point  B ,  whose  co-ordinates  are 
y  =  o,  x  =  +  R,  is  (Art.  60) 

y  =  a(x  —  R); 

and  for  a  line  passing  through  the 
point  B',  for  which  y  =  o  and  x  =  —  R, 
y  =  af  (x  +  R). 

In  order  that  these  lines  should  intersect  on  the  circum 
ference  of  the  circle,  these  equations  must   subsist  at  the 
same  time  with  the  equation  of  the  circle.     Combining  the 
equations  with  that  of  the  circle,  by  multiplying  the  two  first 
together,  and  dividing  by  the  equation  of  the  circle,  we  have 

first 

y2  =  aa'  (x2  —  R2); 

and  the  division  by  if  —  (R2  —  #2),  gives 

aa'  =  —  1,  or  aa'  -f  1  =  o  ; 

but  this  last  equation  expresses  the  condition  that  two  lines 
should  be  perpendicular  to  each  other  (Art.  64) ;  he?ice, 
the  supplemental  chords  of  the  circle  are  perpendicular  to 
each  other. 

138.  The  equation  of  the  circle  may  be  put  under  another 
form,  by  referring  it  to  a  system  of  co-ordinate  axes,  whose 
origin  is  at  the  extremity  B'  of  its  diameter  B'B.  For  any 
point  M,  we  have 

AP  =  x  =  BT  —  B'A  =  x'  —  R. 

Substituting   this  value  of  x  in  the  equation  if  +  x2  =  R2, 

we  get 

if  +  x'2  —  2Ra?'  =  o. 


CHAI.  IV.]  ANALYTICAL  GEOMETRY.  99 

In  this  equation  x'  =  o  gives  y  =  o,  since  the  origin  of  CCH 
ordinates  is  a  point  of  the  curve.  Discussing  this  equation 
as  we  have  done  the  preceding,  we  shall  arrive  at  the  same 
results  as  those  which  have  just  been  determined. 

139.  If  the  circle  be  referred  to  a  system  of  rectangular 
co-ordinates  taken  without  the  circle,  calling  x  and  y'  the 
co-ordinates  of  the  centre,  and  x  and  y  those  of  any  one  of 
its  points,  we  shall  have 

x  —  o?'  =  BC,  y  —  y'=BD; 

and  calling   the   radius   R, 
we  have  (Art.  59), 


which  is  the  most  general 
equation  of  the  circle,  re 
ferred  to  rectangular  axes. 


EXAMPLES. 

1.  Construct  the  equation 

if  +  x*  +  4y  —  4r  —  -8  =  0. 

By  adding  and  subtracting  8,  this  equation  can  be  put 
under  the  form 

if  +  4y  +  4  +  a*  —  4x  +  4—  16  =  o, 
or  (y  +  2)'  +  (a;  —  2)2  =  16. 

Comparing  this  equation  with  that  of  the  general  equation 


we  see  that  it  is  the  equation  of  a  circle,  in  which  the  co 
ordinates  of  the  centre  are  x'  =  2,  y'  =  —  2,  and  whose 
radius  is  4. 


100  ANALYTICAL  GEOMETRY.  [CKAP.  IV 

2.  2y»  +  2o?  —  4y  —  4*  +  1  =  o,  a;'  =  1,  y»  =  1,  R  =  ^7 

3.  2/2  +  z2  —  6*/+4*  —  3=o,    a?'  =—2,  y'  =  3,  R  =  4. 

4.  G^  +  Go;2—  21y—  8a?+14  =  o,  a?'=+f,  y'  =  J,  R  =  f  f. 

5.  jf  +  a^^^y  —  3a?  =  o,    a/  =  f,  y^  =  —  2,  R  =  |. 

6.  t/2  +  a;2  —  4y  =  o,        «'  =  o,     y'  =  2,     R  =  2. 

7.  y2  +  a?2  +  6x  =  o,        x'  =  —  3,    y'  =  o,    R  =  3. 

8.  y2  +  ^2  —  6z  +  8  =  o,        x'  =  3,    y'  =  o,     R  =  1. 

140.  To  find  the  equation  of  a  tangent  line  to  the  circle, 
let  us  resume  the  equation 

a»  +  y"  =  R2. 

Let  x"f  y",  be  the  co-ordinates  of  the  point  of  tangency, 
they  must  satisfy  the  equation  of  the  circle,  and  we  have 

x"2  +  y"2  =  R2. 

The  equation  of  the   tangent   line  will  be  of   the  form 
|.         (Art.  60), 

y  —  y"  =  a  (x  —  x")  ; 

:' 

it  is  required  to  determine  a. 

For  this  purpose,  let  the  tangent  be  regarded  as  a  secant, 
and  let  us  determine  the  co-ordinates  of  the  points  of  inter 
section.  These  co-ordinates  must  satisfy  the  three  preceding 
equations,  since  the  points  to  which  they  belong  are  common 
to  the  line  and  circle.  Combining  these  equations,  by  sub 
tracting  the  second  from  the  first,  we  have 


or  (y  —  y")  (y  +  y")  +  (x  —  x")  x  +  x")  =  o 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  101 

Putting  for  y,  its  value  y"  +  a  (x  —  x")  drawn  from  the 
equation  of  the  line,  we  get 

(y"  +  a  (x  —  x")  —  y")  (y"  +  a  (x  —  x")  +  y")  +  (x  —  x") 
(x  +  x")  =  a  (x  —  x")(2y"+a  (x  —  x")  +  (x  —  x")  (x  +  x") 
=  \2ay"  +  a2  (x  —  x"}  +  x  +  x"|  (x  —  x")  =  o. 

This  equation  will  give  the  two  values  of  x  corresponding 
to  the  two  points  of  intersection.  The  co-ordinates  of  one 
point  are  obtained  by  putting 

x  —  x"  —  o, 
which  gives 

x  =  x",  and  y  =  y"  ; 

and    those  of  the   second    point    are    made   known  by  the 
equation 

Zay"  +  a2  (x  —  x")  +  x  +  x"  =  o, 

when  a  is  given. 

If  now  we  suppose  the  points  of  intersection  to  approach 
each  other,  the  secant  line  will  become  a  tangent,  when 
those  points  coincide;  but  this  supposition  makes 

x  =  x",  and  y  =  y"; 
and  the  last  equation  becomes 

2<n/"  +  2x"  =  o, 
from  which  we  get 


Substituting  this  value  of  a  in  the  equation  of  the  tangent, 
it  becomes 


hence  yy"  -f  xx'  =  Rs, 

which  is  the  equation  of  a  tangent  line  to  the  circle. 
9* 


102  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

Putting  it  under  the  form 

x"          W 

y  =  --  17  *  -\  --  r,  .• 

y        y 

and  comparing  this  equation  with  that  of  the  straight  line  in 

x" 

Art.  52,  we  see  that  --  -,  is  the  tangent  of  the  angle  which 
u 

the  tangent  line  makes  with  the  axis  of  x. 

The  value  which  we  have  just  found  for  a  being  single,  it 
fo\\bvfs  thrt'but  one  tangent  can  be  drawn  to  the  circle,  at  a 
given  point  of  the  curve. 

141.  A  line  -drawn  through  the  point  of  tangency  perpen 
dicular  to  the  tangent  is  called  a  Normal.  Its  equation  will 
be  of  the  form 

y  —  y"  =  a'  (x  —  x"). 

The  condition  of  its  being  perpendicular  to  the  tangent 
gives 

da  +  l  =  o,  or  a'  =  --  . 

a 

But  we  have  found  (Art.  140), 

x" 


hence, 


x" 


Substituting  this  value  in  the  equation  of  the  normal,  it 
becomes 


and  reducing,  we  have 

yx"  —  y"x  =  o, 
for  the  equation  of  the  normal  line  to  the  circle. 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  103 

14*2.  The  normal  line  to  the  circle  passes  through  its  centre, 
which,  in  this  case,  is  the  origin  of  co-ordinates.  For,  if  we 
make  one  of  the  variables  equal  to  zero,  the  other  will  be 
zero  also.  Hence  the  tangent  to  a  circle  is  perpendicular  to 
the  radius  drawn  through  the  point  of  tangency. 

143.  To  draw  a  tangent  to  the   circle,  through  a  point 
without  the  circle,  let  x  y'  be  the  co-ordinates  of  this  point. 
Since  it  must  be  on  the  tangent,  it  must  satisfy  the  equation 
of  this  line,  and  we  have  eq.  of  tangent  yyff  -f-  xx"  =  R2 

y'  y"  +  x'  x"  =  R2. 
We  have  besides, 

y"2  +  x"2  =  R2. 

These  two  equations  will  determine  x'  and  y",  the  co-or 
dinates  of  the  point  of  tangency,  in  terms  of  R  and  the  co 
ordinates  x  y'  of  the  given  point.  Substituting  these  values 
in  the  equation  of  the  tangent,  it  will  be  determined. 

The  preceding  equations  being  of  the  second  degree,  will 
give  two  values  for  x"  and  y".  There  will  result  conse 
quently  two  points  of  tangency,  and  hence  two  tangents 
may  be  drawn  to  a  circle  from  a  given  point  without  the 
circle. 

144.  We  have  seen  that  the  equation  of  the  circle  referred 
to  rectangular  co-ordinates,  having  their  origin  at  the  centre, 
only  contains  the  squares  of  the  variables  x  and  ?/,  and  is  of 
the  form 

if  +  **  =  R'. 

Let  us  seek  if  there  be  any  other  systems  of  axes,  to 
which,  if  the  curve  be  referred,  its  equation  will  retain  the 
5ame  form. 

Let  us  refer  the  equation  of  the  circle  to  systems  having 


104  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

the  same  origin,  and  whose  co-ordinates  are  represented  by 
x'  and  y.  Let  a,  a',  be  the  angles  which  these  new  axes 
make  with  the  axis  of  x.  We  have  for  the  formulas  of  trans 
formation  (Art.  Ill), 

x  —  x'  cos  a  +  y'  cos  a',         y  =  x'  sin  a  +  y'  sin  a'. 

Substituting  these  values  for  x  and  y  in  the  equation  of 
the  circle,  it  becomes 

y'2  (cos  V  +  sin  V)  +  2a?y  cos  (a'  —  a)  +  x'2 

(cos2a  +  sin2a)  =  R2; 
or,  reducing, 

y'2  +  2x'y'  cos  (a  — -  a)  +  x'2  =  R'. 

The  form  of  this  equation  differs  from  that  of  the  given 
equation,  since  it  contains  a  term  in  x'y'.  In  order  that  this 
term  disappear,  it  is  necessary  that  the  angles  a  a'  be  such 
that  we  have 

COS  (a'  —  a)  =  0, 

which  gives  (a'  —  a)  =  90°,  or  270°  ; 
hence  a'  =  a  +  90°,  or  a'  =  a  +  270°, 

which  shows  that  the  new  axes  must  be  perpendicular  to 
each  other. 

145.  Conjugate  Diameters  are  those  diameters  to  which,  if 
the  equation  of  the  curve  be  referred,  it  will  contain  only  the 
square  powers  of  the  variables.  In  the  circle,  we  see  that 
these  diameters  are  always  at  right  angles  to  each  other;  and 
as  an  infinite  number  of  diameters  may  be  drawn  in  the 
circle  perpendicular  to  each  other,  it  follows  that  there  will 
be  an  infinite  number  of  conjugate  diameters. 


CHAP.  IV.] 


ANALYTICAL  GEOMETRY. 


305 


Of  the  Polar  Equation  of  the  Circle. 

146.  To  find  the  equation  of  the  circle  referred  to  polar 
co-ordinates,   let   O    be    taken 

as  the  pole,  the  co-ordinates  of 
which  referred  to  rectangular 
axes  are  a  and  b;  draw  OX' 
making  any  angle  a  with  the  axis 
of  x.  OM  will  be  the  radius 
vector,  and  MOX'  the  variable 
angle  v.  The  formulas  for  trans 
formation  are  (Art.  123), 

x  =  a  +  r  cos  (v  +  a),     y  =  b  +  r  sin  (v  +  a). 
These  values  being  substituted  in  the  equation  of  the  circle 

y2  +  tf  =  R2, 
it  becomes 

r2  +  2  ja  cos  (v  +  «)  +  b  sin  (v  +  a)  j  r  +  a*  +  52  —  R1  =  o. 

which  is  the  most  general  polar  equation  of  the  circle. 

This  equation  being  of  the  second  degree  with  respect  to 
r,  will  generally  give  two  values  to  the  radius  vector.  The 
positive  values  alone  must  be  considered,  as  the  negative 
values  indicate  points  which  do  not  exist  (Art.  124). 

147.  By  varying  the  position  of  the  pole  and  the  angle  v, 
this  equation  will  define  the  position  of  every  point  of  the 
circle. 

If  the  pole  be  taken  on  the  circumference,  and  we  call  a, 
b,  its  co-ordinates,  these  co-ordinates  must  satisfy  the  equation 
of  the  circle,  and  we  have  the  relation 


106  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

The  polar  equation  reduces  to 

r2  +  2  ja  cos  (v  +  a)  +  b  sin  (v  +  a)  r  j  =  o. 

If  OX'  be  parallel  to  the  axis  of  x,  the  angle  a  will  be  zero, 
and  this  equation  becomes 

r2  +  2  (a  cos  v  +  b  sin  v)  r  =  o. 

This  equation  may  be  satisfied  by  making  r  =  o.  Hence, 
one  of  the  values  of  the  radius  vector  is  always  zero,  and  it 
may  be  satisfied  by  making 

r  4-  2  (a  cos  v  +  b  sin  v)  =  o, 
which  gives 

r  =  —  2  (a  cos  v  +  b  sin  v) ; 

from  which  we  may  deduce  a  second  value  for  the  radius 
vector  for  every  value  of  the  angle  v. 

148.  If  we  have  in  this  last  equation  r  =  o,  the  equation 

becomes 

a  cos  v  +  b  sin  v  =  o, 

sinu  a 

or = r » 

cosv  o 

a 
or  tang  v  =  —  -7- » 

a  relation  which  has  been  before  obtained  (Art.  140). 

149.  If  the  pole  be  taken  at  the  centre  of  the  circle,  a  and 
b  would  be  zero,  and  the  formulas  for  transformation  would  be 

x  =  r  cos  v,     y  =  r  sin  v. 

Of  the  Ellipse. 

150.  We  have  found  (Art.  127,)  for  the  general  equation 
of  intersection  of  the  cone  and  plane, 

if  tang  *v  +  x*  cos  *u  (tang  *v  —  tang  *u)  +  2  ex  sin  u  =  <?, 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  107 

and  that  this  equation  represents  a  class  of  curves  called 
Ellipses,  when  u  <  v.  We  will  now  examine  their  peculiar 
properties. 

To  facilitate  the  discussion,  let  us  transfer  the  origin  of 
co-ordinates  to  the  vertex  B  of  the  curve. 
For  any  abscissa  OP'  =  x,  we  wrould  have 

*  =  OB  —  BP; 
or  calling  the  new  abscissas  x'f 

x  =  OB  —  x',  and  y  =  y '. 
But  in  the  triangle  BOG  we 
have  the  angle  C  =  90°  —  v, 
and  the  angle  B  —  v  +  u  and 
the  side  OC  =  c,  and  we  get 
C  sin  OCB 


OB  = 


sin  (v  -f-  u) 

C  COS  V 


sin  (v  +  u) 

C  COS  V 


sin  (v  - 
from  which  results 


c  cos  v 


x  = 


—  x' 


sin  (y  +  u) 

Substituting  this  value  of  x 
in  the  equation  of  the  curve, 
we  have 

t/'2  sin  *v  +  x'2  sin  (v  +  u)  sin  (u  —  u)  —  2cx'  sin  v 
cos  v  cos  u  =  o  ; 

and  suppressing  the  accents,  we  have 

y2  sin  zv  +  x2  sin  (v  +  u)  sin  (v  —  u)  —  2c#  sin  v 
cos  v  cos  u  —  o; 

which  is  the  general  equation  of  the  intersection  of  the  cone 
and  plane,  referred  to  the  vertex  B. 


108  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

151.  To  discuss  this  equation  when  u  <^vt  let  us  first  find 
the  points  in  which  it  meets  the  axis  of  x.  Making  y  =  o, 
we  have 

x9  sin  (v  +  u)  sin  (v  —  u)  —  2cx  sin  v  cos  v  cos  u  =  o; 
which  gives  for  the  two  values  of  x, 

2c  sin  v  cos  v  cos  u 

x  =  o,  and  x  =  — — - — • — r~ - — > 

sin  (v  -f-  u)  sin  (v  —  u) 

which    shows    that  it  cuts    the    axis  of  x  in  two  points  B 
and    B',   one    at    the    origin,   the    other   at    the    distance 

2ca?  sin  v  cos  v  cos  u 

-. — ; — : — -. r  on  the  positive  side  of  the  axis  of  y. 

sin  (v  +  u)  sin  (v  —  u) 

Making  x  =  o,  we  have  the  points 
)B      in  which  it  cuts  the  axis  of  y.    This 
supposition  gives 

*?  =  o, 

which  shows  that  the  axis  of  y  is  tangent  to  the  curve  at  B, 
the  origin  of  co-ordinates. 

Resolving  this  equation  with  respect  to  y,  we  have 

y=  

rh  — \  / — x2  sin  (v  +  u)  sin  (v — u)  +  %cx  sin  v  cos  u  cos  v. 

sin  v  V 

These  two  values  being  equal,  and  with  contrary  signs, 
the  curve  is  symmetrical  with  respect  to  the  axis  of  x.  If 
we  suppose  x  negative,  y  becomes  imaginary,  since  this  sup 
position  makes  all  the  terms  under  the  radical  essentially 
negative.  The  curve,  therefore,  is  limited  in  the  direction 
of  the  negative  abscissas.  If,  on  the  contrary,  we  suppose  r 
positive,  the  values  of  y  will  be  real,  so  long  as 

a?  sin  (v  +  u)  sin  (v  —  u)  <  %cx  sin  v  cos  v  cos  u, 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  109 

cr, 

2c  sin  v  cos  v  cos  u 

^  sin  (v  +  u)  sin  (v  —  u)  ' 

and  they  become  imaginary  beyond  this  limit.     The  curve, 
therefore,  extends  from  the  origin  of  co-ordinates  a  distance 

2c  sin  v  cos  v  cos  u  .  .         . 

BB  =  —. — r— — ; r   on    the   positive   side   of  the 

sin  (v  +  u)  sin  (u  —  u) 

axis  of  x. 

Let  us  refer  the  curve  to  the 
point  A,  the  middle  of  BB'.  The 
formula  for  transformation  will 
be,  for  any  point  P,  BC  =  AB 
—  AC,  or  calling  BC,  x,  and  AC,  x', 

c  sin  v  cos  v  cos  u 

x  =  —. x'» 

sin  (v  +  u)  sin  (v  —  u) 

Substituting  this  value  in  the  equation  of  the  ellipse, 
\f  sin  *v  -f  x2  sin  (u  +  u)  sin  (v  —  u)  —  2cx  sin  v  cos  v  cos  u  =  o, 

and  reducing,  we  have 

.  .   .  2   •    /          \    •     /  c2sin2ucos2t-cos2M 

y*s\n*v  +  x   sin  (v  +  u)  sin  (v — u)  —  - 


sin  (»+tt)  sin  (v — it) 
which  is  the  equation  of  the  ellipse  referred  to  the  .point  A. 

Making  y  —  o,  we  find  the  abscissas  of  the  points  B  and  B', 
in  which  the  curve  cuts  the  axis  of  x. 

c  sin  v  cos  v  cos  u 

sin  (v  +  u)  sin  (i;  —  u) ' 

c  sin  v  cos  v  cos  u 

sin  (v  +  u)  sin  (v  —  u)  ' 

and  x  =  o  gives  the  ordinates  AD  and  AD'. 

c  cos  v  cos  u 
•/sin  (v  +  u)  sin  (v  —  u) 
10 


110  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

152.  This  equation  takes  a  very  simple  and  elegant  form 
when  we  introduce  in  it  the  co-ordinates  of  the  points  in 
which  the  curve  cuts  the  axes.  For,  if  we  suppose 

c2  sin  2v  cos  2v  cos  2u 

A"  =  -.— 2-, — ; — v — =— r; r »  and 

sin  (v  +  u)  sin  (v  —  u) 

cz  cos  2v  cos  2w 


sin  (o  +  u)  sin  (v  —  u)  ' 

we  have  only  to  multiply  all  the  terms  of  the  equation  in  y 
and  x'9  by 

c2  cos  2v  cos  2u 
sin2  (v  +  u)  sin2  (v  —  u)  ' 

and  putting  x  for  a?',  we  have 

,       c2  sin  \>  cos  2v  cos  2w  c2  cos  2v  cos  2w 

^.2 I       ,jj , __ 

y  sin2  (v  +  M)  sin2  (u  —  u)  sin  (*;  +  u)  sin  (u  —  w) 

c2  sin2y  cos  2vcos2w  c2  cos  2u  cos  a& 

sin2  (v  +  u)  sin2  (v  —  u)       sin  (u  +  w)  sin  (v  —  u)  ' 

and  making  the  necessary  substitutions,  we  obtain 

Ay  +  B  V  -  A2B2. 

The  quantities  2A  and  2B  are  called  the  Axes  of  the  Ellipse. 
2A  is  the  greater  or  transverse  axis ;  2B  the  conjugate  or  less 
axis.  The  point  A  is  the  centre  of  the  ellipse,  and  the 

equation 

Ay  +  BV  =  A2B2 

is  therefore  the  equation  of  the  Ellipse  referred  to  its  centre 
and  axes. 

153.  If  the  axes  are  equal  we  have  A  =  B,  and  the  equa 
tion  reduces  to 

which  is  the  equation  of  the  circle. 


CHAP.  IV.J  ANALYTICAL  GEOMETRY.  Ill 

154.  Every  line  drawn  through  the  centre  of  the  ellipse  is 
called  a  Diameter,  and  since  the  curve  is  symmetrical,  it  is 
easy  to  see  that  every  diameter  is  bisected  at  the  centre. 

2B2 

155.  The  quantity   — r-    is   called    the    parameter  of  the 

J\. 

curve,  and  since  we  have 

2B2 
2A  :  2B  :  :  2B  :  -r- , 

A. 

it  follows  that  the  parameter  of  the  ellipse  is  a  third  propor 
tional  to  the  two  axes. 

156.  Introducing  the  expressions  of  the  semi-axes  A  and 
B  in  the  equation 

y*  sin  2v  +  x2  sin  (v  +  u)  sin  (v  —  u)  — 2c#  sin  v 
cos  v  cos  u  =  o, 

in  which  the  origin  is  at  the  extremity  of  the  transverse  axis, 
by  multiplying  each  term  by  the  quantity. 

,£  2  2 

sin2  (v  +  u)  sin2  (v  —  u) ' 
it  becomes  • 

AV  +  BV  — 


which  may  be  put  under  the  form 


If  we  designate  by  x',  y  ,  x",  y"9  the  co-ordinates  of  anv 
two  points  of  the  ellipse,  we  shall  have 

y2  _  x'  (2A  -  x') 
y'2       x'  (2  A—  x")' 

which  shows  that  in  the  'ellipse,  the  squares  of  the  ordinates 
are  to  each  other  as  the  products  of  the  distances  from  the  foot 
of  each  ordinate  to  the  vertices  of  the  curve. 


112  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

157.  The  equation  of  the  ellipse  referred  to  its  centre  and 
axes  may  be  put  under  the  form 

If  from  the  point  A  as  a 
centre  with  a  radius  AB  =  A, 
we  describe  a  circumference 
of  a  circle,  its  equation  will 
be 

7/    zi^    A     i- .  _  _  *Y*^ 

Representing  by  y  and  Y  the  ordinates  of  the  ellipse  and 
circle,  which  correspond  to  the  same  abscissa,  we  have,  by 
comparing  these  two  equations. 


According  as  B  is  less  or  greater  than  A,  y  will  be  less 
or  greater  than  Y,  hence  if  from  the  centre  of  the  ellipse  with 
radii  equal  to  each  of  its  axes,  two  circles  be  described)  the 
ellipse  will  include  the  smaller  and  be  inscribed  within  the 
large  circle. 

158.  From  this  property  we  deduce,  1st.  That  the  trans 
verse  axis  is  the  longest  diameter,  and  the  conjugate  the 
shortest;  2dly.  When  we  have  the  ordinates  of  the  circle 
described  on  one  of  the  axes,  to  find  those  of  the  ellipse,  we 
have  only  to  augment  or  diminish  the  former  in  the  ratio  of 
B  to  A.  This  gives  a  method  of  describing  the  ellipse  by 
points  when  the  axes  are  known. 

From  the  point  A  as  a  centre  with  radii  equal  to  the  semi- 
axes  A  and  B,  describe  the  circumferences  of  two  circles, 
draw  any  radius  ANM,  and  through  M  draw  MP  perpen- 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  113 

dicular  to  AB,  and  through  N  draw  NQ  parallel  to  AB.    The 
point  Q  will  be  on  the  ellipse,  for  we  have 


or, 


as  in  Art.  157. 

159.  We  have   seen    that  for  every  point  on  the  ellipse, 
the  value  of  the  ordinate  is 

y2  — /A2 yf\t 

For  a  point  without  the  ellipse,  the  value  of  y  would  be 
greater  for  the  same  value  of  x,  and  for  a  point  within,  the 
value  of  y  would  be  less.  Hence, 

For  points  without  the  ellipse,        Ay  +  BV  — A2B2>o. 
For  points  on  the  ellipse,  Ay  +  BV  —  A2B2  =  o. 

For  points  within  the  ellipse,          Ay  +  BV  —  A2B2  <  o. 

160.  If  through  the  point  B',  whose  co-ordinates  are  y  =  o 
*=  —  A,  we  draw  a  line,  its  equation  will  be 

y  =  a  (x  +  A). 

For  a  line  passing  through  B, 
whose  co-ordinates  are  y  =  o, 
x  =  +  A,  we  have 


y  =  a'  (x  —  A.) 

If  it   be  required   that    these 
lines  should  intersect  on  the  el 
lipse,  it  is  necessary  that  these  equations  subsist  at  the  same 
10*  P 


114  ANALYTICAL  GEOMETRY.  [CHAP.  W. 

time  with  the  equation  of  the  ellipse.      Multiplying  them 
together,  we  have 


and  in  order  that  this  equation  agree  with  that  of  the  ellipse, 

B2 
tf^p-A 

we  must  have 

B2  B2 

—  aa  ==  A*  '  or  aa'  =  —  —  , 

which  establishes  a  constant  relation  between  the  tangents  of 
angles  formed  by  the  chords  drawn  from  the  extremities  of 
the  transverse  axis  with  this  axis.  In  the  circle  B  =  A,  and 
this  relation  becomes 

aa'  =  —  1, 

as  we  have  seen  (Art.  137). 

161.  When  the  relation  which  has  just  been  established 
(Art.  160)  takes  place  between  the  angles  which  any  two 

lines  form  with  the  axis  of  x,  these  lines  are  supplementary 

j^ 
chords  of  an  ellipse,  the  ratio  of  whose  axes  is  ^> 

162.  As  we  proceed  in  the  examination  of  the  properties 
of  the  ellipse,  we  are  struck  with  the  great  analogy  between 
this  curve  and  the  circle.    We  may  trace  this  analogy  farther. 
In  the  circle  we  have  seen  that  all  the  points  of  its  circum 
ference  are  equally  distant  from  the  centre.     Although  this 
property  does  not  exist  in  the  ellipse,  we  find  something  ana 
logous  to  it  ;  for,  if  on  the  transverse  axis  we  take  two  points 


F,  F',  whose  abscissas  are  ±  VA2  —  B^,  the  sum  of  the  dis 
tances  of  these  points  to  the  same  point  of  the  curve  is  al 
ways  constant  and  equal  to  the  transverse  axis.  To  prove 


CHAP.  1VY1 


ANALYTICAL  GEOMETRY. 


115 


this,  let  x  and  y  be  the  co-ordinates  of  any  point  M  of  the 
ellipse;  represent  the  abscissas  of  the  points  F,  F'  by 


Calling  D  the  distance  MF,  or  MF',  we  have  (Art.  59), 


but  since 
we  have 


y'  =  o, 
D2  =  z/2  +  (x  —  x')z. 


Putting  for  y  its  value  drawn  from  the  equation  of  the 
ellipse,  and  substituting  for  x'2  its  value  A2  —  B2,  this  expres 
sion  becomes 

D2  =  B2  —  ^  +  x*  —  2xx'  +  A2  —  B2  = 


— ^2—  or2  — 2*0;'  +  A2; 
or,  substituting  for  A2  —  B2  its  value  a:'2, 


Extracting  the  square  root  of  both  members,  we  have 


Taking  the  positive  sign,  and  substituting  for  x'  its  two 
values  =fc  x/A2  — B2,  we  have  for  the  distance  MF,  or  MFr, 


11G 


ANALYTICAL  GEOMETRY. 


[CHAP.  IV. 


MF  =  A  — 


—  B2 


MF'  =  A  + 


x 


—  B 


A  A 

Adding  these  values  together,  we  get 
MF  +  MF'  =  2A, 

which  proves  that  the  sum  of  the  distances  of  any  point  of  the 
ellipse  to  the  points  F,  F',  is  constant  and  equal  to  the  trans 
verse  axis. 

163.  The  points  F,  F',  are  called  the  Foci  of  the  ellipse, 
and  their  distance  ±\/A2 —  B2  to  the  centre  of  the  ellipse 
is  called  the  Eccentricity.  When  A  =  B,  the  eccentricity 
=  o.  The  foci  in  this  case  unite  at  the  centre,  and  the  ellipse 
becomes  a  circle.  The  maximum  value  of  the  eccentricity  is 
when  it  is  equal  to  the  semi-transverse  axis.  In  this  suppo 
sition  B  =  Of  and  the  ellipse  becomes  a  right  line. 


Making   a?  =  rfc  v/A2—  B2   in  the  equation  of  the  ellipse, 

we  find 

B2  2B2 


which  proves  that  the  double  ordinate  passing  through  the 

focus  is  equal  to  the  parameter. 

164.  The   property   demonstrated  (Art.  162)  leads    to   a 

very  simple  construction  for  the  ellipse.     From  the  point  B 

lay  off  any  distance  BK  on  the 
axis  BB'.  From  the  point  F  as 
a  centre,  with  a  radius  equal  to 
BK,  describe  an  arc  of  a  circle  ; 
and  from  F'  as  a  centre,  with  a 

radius  B'K,  describe  another  arc.     The  point  M  where  these 

arcs  intersect,  is  a  point  of  the  ellipse.     For 
MF  +  MF'  =  2A. 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  117 

When  we  wish  to  describe  the  ellipse  mechanically,  we 
fix  the  extremities  of  a  chord  whose  length  is  equal  to  the 
transverse  axis,  at  the  foci,  F,  F',  and  stretch  it  by  means  of 
a  pin,  wrhich  as  it  moves  around  describes  the  ellipse. 

165.  To  find  the  equation  of  a  tangent  line  to  the  ellipse, 
let  us  resume  its  equation, 

Ay  +  BV  =  A2B2. 

Let  x",  y1',  be  the  co-ordinates  of  the  point  of  tangency, 
they  will  verify  the  relation, 

Ay/2  +  BV'a  =  AsBa. 

The  tangent  line  passing  through  this  point,  its  equation 
will  be  of  the  form 

y  —  y"  =  a(x  —  x"). 

It  is  required  to  determine  a. 

To  do  this,  we  will  find  the  points  in  which  this  line  con 
sidered  as  a  secant  meets  the  curve.  For  these  points  the 
three  preceding  equations  must  subsist  at  the  same  time. 
Subtracting  the  two  first  from  each  other,  wre  have 

^  (y — y"}  (y  +  y")  +  B2  (x — x")  (x  +  x")  =  o. 

Putting  for  y  its  value  y"  +  a  (x  —  x")  drawn  from  the 
equation  of  the  line,  w*e  find 

(x  —  x")  I  A2  (2ay"  +  «2  (x  —  x") )  +  B2  (x  +  x")  \  =  o 
This  equation  may  be  satisfied  by  making 

x  —  x"  =  o, 
which  gives 

•  »*•, 

from  which  we  get 


118  ANALYTICAL  GEOMETR^.  [CHAP.  IV 

and  also  by  making 

A*  \2ay"  +  a*  (a?  —  a?")!  +  B2  (x  +  x")  =  o. 

Now  when  the  secant  becomes  a  tangent,  we  must  have 
x  =  x",  which  gives 

Aaay"  +  BV  =  0/ 

hence 

BV 

~ 


Substituting  this  value  of  a  in  the  equation  of  the  tangent, 
it  becomes 

BV 

y-y"    -(x~afl)i 


or  reducing,  and  recollecting  that  A2?/"2  +  BV2  —  A2B2,  we 
have 

A*yy"  +  B*xx"  =  A2  B2 

for  the  equation  of  the  tangent  line  to  the  ellipse. 

166.  If  through  the  centre  and  the  point  of  tangency  we 
draw  a  diameter,  its  equation  will  be  of  the  form 

y'  =  a1  x", 
from  which  we  get 


But  we  have  just  found  the  value  of  a,  corresponding  to 
the  tangent  line,  to  be 

BV 

ft   -   ~~~ 


Multiplying  these  values  of  a  and  a    together,  we  find 

B2 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  119 

This  relation  being  the  same  as  that  found  in  Arts.  160,  161, 
shows  that  the  tangent  and  the  diameter  passing  through  the 
point  of  tangency,  have  the  property  of  being  the  supple 
mentary  chords  of  an  ellipse,  whose  axes  have  the  same 

A 
ratio  ^-' 

£> 

167.  This  furnishes  a  very  simple  method  of  determining 
the  direction  of  the  tangent.  For  if  we  draw  any  two  sup 
plementary  chords,  and  designate  by  a,  «'  ',  the  trigonometri 
cal  tangents  of  the  angles  which  they  make  with  the  axis, 
we  have  always  between  them  the  relation 


We  may  draw  one  of  these  chords  parallel  to  the  diameter, 
passing  through  the  point  of  tangency.     In  this  case  we  have 

a  =a! 
from  which  results  also 

a  =  a; 

that  is,  the  other  chord  will  be  parallel  to  the  tangent. 
168.  To  draw  a  tangent  through  a  point  M  taken  on  the 


ellipse,  draw  through  this  point  AM,  and  through  the  ex- 
'•remity  B'  of  the  axis  BB'  draw  the  chord  B'N  parallel  to 


120  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

AM ;  MT  parallel  to  BN  will  be  the  tangent  required.  We 
see,  by  this  construction  .also,  that  if  we  draw  the  diameter 
AM'  parallel  to  the  chord  BN,  or  to  the  tangent  MT,  the 
tangent  at  the  point  M'  will  be  parallel  to  the  chord  B'N,  or 
to  the  diameter  AM. 

169.  When  two  diameters  are  so  disposed  that  the  tangent 
drawn  at  the  extremity  of  one  is  parallel  to  the  other,  they 
are  called  Conjugate  Diameters.     It  will  be  shown  presently 
that  these  diameters  enjoy  the  same  property  in  the  ellipse 
as  those  demonstrated  for  the  circle  (Arts.  144,  145). 

170.  To  find  the  subtangent  for  the  ellipse,  make  y  =  o  in 
the  equation  of  the  tangent  line. 

A*yy"  +  Wxx"  =  A2B2, 

we  have  for  the  abscissa  of  the  point  in  which  the  tangent 
meets  the  axis  of  a:, 

A2 

x  =  —  > 
x1 

which  is  the  value  of  AT.  If  we  subtract  from  this  ex 
pression  AP  =  x" ',  we  shall  have  the  distance  PT,  from  the 
foot  of  the  ordinate  to  the  point  in  which  the  tangent  meets 
the  axis  of  x.  This  distance  is  called  the  subtangent.  Its 
expression  is 

A  2  ,y,"2 

PT  =      ""—  • 

x" 

This  value  being  independent  of  the  axis  B,  suits  every 
ellipse  whose  semi-transverse  axis  is  A,  and  which  is  con 
centric  with  the  one  we  are  considering.  It  therefore  cor 
responds  to  the  circle,  described  from  the  centre  of  this 
ellipse  with  a  radius  equal  to  A.  Hence,  extending  the 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  121 

ordinate  MP,  until  it  meets 

the  circle  at  M',  and  draw 

ing  through  this   point  the 

tangent  M'T,  MT  will    be 

tangent  to  the  ellipse  at  the 

point  M.    This  construction 

applies  equally  to  the  conjugate  axis,  on  which  the  expression 

for  the  subtangent  would  be  independent  of  A. 

171.   To  find  the  equation  of  a  normal  to  the  ellipse,  its 
equation  will  be  of  the  form 

y  —  y"  =  a'(x  —  x"). 

The  condition  of  its  being  perpendicular  to  the  tangent, 
for  which  we  have  (Art.  165), 

BV 

"AY  ' 

requires  that  there  exist  between  a  and  a  the  condition 

aa   -r  1  =  o, 
which  gives 

a>-^. 
BV 

This  value  being  substituted  in  the  equation  for  the  normal, 
gives 


172.  To  find  the  subnormal  for  the  ellipse,  make  y  =  o  in 
the  equation  of  the  normal,  and  we  have  for  the  abscissa  of 
the  point  in  which  the  normal  meets  the  axis  of  x, 


11 


122  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

This  is  the  value  of  AN.  Subtracting  it  from  AP,  which 
is  represented  by  x",  we  shall  have  the  distance  from  the 
foot  of  the  ordinate  to  the  foot  of  the  normal.  This  distance 
is  the  subnormal,  and  its  value  is  found  to  be 

"RV 

°  x 


173.  The  equation  of  the  ellipse  being  symmetrical  with 
respect  to  its  axes,  the  properties  which  have  just  been  de 
monstrated  for  the  transverse,  will  be  found  applicable  also 
to  the  conjugate  axis. 

174.  The  directions  of  the   tangent   and   normal  in  the 
ellipse  have  a  remarkable  relation  with  those  of  the  lines, 
drawn  from  the  two  foci  to  the  point  of  tangency.     If  from 
the  focus  F,  for  which  y  =  o  and  x  =  VA2 — B2,  we  draw 

v  a  straight  line  to  the 

3f  point  of  tangency,  its 

equation  will  be  of  the 
form 

y  —  y"  —  a.  (x  —  x"). 


If  we  make  for  more  simplicity  VA2 —  B2  =  c,  the  con 
dition  of  passing  through  the  focus  will  give 


hence, 


—  y    =  a  (C  — 

—A 


But  we  have  for  the  trigonometrical  tangent  which  the 
tangent  line  makes  with  the  axis  of  x  (Art.  165), 

BV 
AV* 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  123 

The  angle  FMT  which  the  tangent  makes  with  the  line 
drawn  from  the  focus  will  have  for  a  trigonometrical 
tangent  (Art.  64), 


+  aa 
Putting  for  a  and  a  their  values,  it  reduces  to 

A2*/'2  +  BV2  —  E2cx" 
A2cy"—  (A2—  B2)x"y"' 

which  reduces  to 

Ba 

w 

in  observing  that  the  point  of  tangency  is  on  the  ellipse,  and 
that  A2—  B2  =  c*. 

In  the  same  manner  the  equation  of  a  line  through  the 
focus  F'  is  found  by  making  x  =  — c,  and  y  =  o  in  the 
equation 

y— y"  =  «'(*  —  *")> 

and  we  have 

—  y"  =  a"(—  C  —  X"\ 

hence 

«'  =  -?- 

c  -f  x 

The  angle  F'MT  w7hich  this  line  makes  with  the  tangent, 
will  have  for  a  trigonometrical  tangent, 

a  —  a!  JB*_ 

1  -4-  aa!  cy" 

when  we  put  for  a  and  a'  their  values. 

The  angles  FMT,  F'MT,  having  their  trigonometrical  tan 
gents  equal,  and  with  contrary  signs,  are  supplements  of  each 
other,  hence 

FMT  +  F'MT  =  180° ; 


124 
but 

hence 


ANALYTICAL  GEOMETRY. 


F'MT  +  F'M*  -  180°, 


[CHAP.  IV 


FMT  =  F'Mf, 

which  shows  that  in  the  ellipse,  the  lines  drawn  from  the  foci 
to  the  point  of  tangency,  make  equal  angles  with  the  tangent ; 
and  it  follows  from  this,  that  the  normal  bisects  the  angle 
formed  by  the  lines  drawn  from  the  point  to  the  same  point  of 
the  curve. 

175.  The   property   just   demonstrated,  furnishes   a  very 

simple  construction  for 
drawing  a  tangent  line 
to  the  ellipse  through  a 
given  point.  Let  M  be 
the  point  at  which  the 
tangent  is  to  be  drawn. 
Draw  FM,  F'M,  and  pro 
duce  F'M  a  quantity  MK 
—  FM.  Joining  K  and 
F,  the  line  MT,  perpen 
dicular  to  FK,  will  be  the  tangent  required ;  for  from  this 
construction,  the  angles  TMF,  TMK,  F'Mf,  are  equal  to 
each  other. 

We  may  see  that  the  line  MT  has  no  other  point  common 
besides  M,  since  for  any  point  t, 

Ft  _f_  F't  >  F'MK  >  2A. 

If  the  given  point  be  without  the  ellipse,  as  at  t,  then 
from  the  point  F'  as  a  centre,  with  a  radius  F'K  =  2A  de 
scribe  an  arc  of  a  circle;  from  the  point  t  as  a  centre,  with 
a  radius  tF,  describe  another  arc,  cutting  the  first  in  K, 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  125 

Drawing  F'K,  the  point  M  will  be  the  point  of  tangency, 
and  joining  M  and  t,  Mt  will  be  the  tangent  required.  For, 
from  the  construction,  we  have  tF  =  tK.  Besides  F'M  + 
FM  =  2A  and  F'M  +  MK  =  2A.  Hence 

MF  =  MK. 

The  .ine  Mt  is  then  perpendicular  at  the  middle  of  FK. 
The  angles  FMT,  F'M*  are  then  equal,  and  *MT  is  tangent 
to  the  ellipse. 

The  circles  described  from  the  points  F'  and  t  as  centres, 
cutting  each  other  in  two  points,  two  tangents  may  be  drawn 
from  the  point  t  to  the  ellipse. 


Of  the  Ellipse  referred  to  its  Conjugate  Diameters. 

176.  There  is  an  infinite  number  of  systems  of  oblique 
axes,  to  which,  if  the  equation  of  the  ellipse  be  referred,  it 
will  contain  only  the  square  powers  of  the  variables.  Sup 
posing  in  the  first  place,  that  its  equation  admits  of  this  re 
duction,  it  is  easy  to  see  that  the  origin  of  the  system  must 
be  at  the  centre  of  the  ellipse.  For,  if  we  consider  any 
point  of  the  curve,  whose  co-ordinates  are  expressed  by 
+  x't  +  ?/',  since  the  transformed  equation  must  contain  only 
the  squares  of  these  variables,  it  is  evident  it  will  be  satisfied 
by  the  points  whose  co-ordinates  are  +  x',  —  y'  ;  — a?',  +  y' ; 
that  is,  by  the  points  which  are  symmetrically  situated  m 
the  four  angles  of  the  co-ordinate  axes.  Hence  every  line 
drawn  though  this  origin  will  be  bisected  at  this  point,  a 
property  which,  in  the  ellipse,  belongs  only  to  its  centre, 
since  it  is  the  only  point  around  which  it  is  symmetrically 
disposed. 

11* 


126  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

The  oblique  axes  here  supposed  will  always  cut  the  ellipse 
in  two  diameters,  which  will  make  such  an  angle  with  each 
other  as  to  produce  the  required  reduction  These  lines  are 
called  Conjugate  Diameters,  which,  besides  the  geometrical 
property  mentioned  in  Art.  169,  possess  the  analytical 
property  of  reducing  the  equation  of  the  curve  to  those  terms 
which  contain  only  the  square  powers  of  the  variables. 

177.  The  equation  of  the  ellipse  referred  to  its  centre  and 
axes  is 

Ay  -f  BV  -  A2B2. 

To  ascertain  whether  the  ellipse  has  many  systems  of  con 
jugate  diameters,  let  us  refer  this  equation  to  a  system  of 
oblique  co-ordinates,  having  its  origin  at  the  centre.  The 
formulas  for  transformation  are  (Art.  Ill), 

x  =  x'  cos  a  +  y'  cos  a',         y  =  x'  sin  a  +  y'  sin  a'. 

Substituting  these  values  for  x  and  y  in  the  equation  of  the 
ellipse,  it  becomes 

c  (A2 sin V  +  B2  cos  V)  y'2  +  (A2  sin2*  +  B2 cos  V)  i 
\     x'2  +  2  (A2  sin  a  sin  a'  +  Bz  cos  a  cos  a')  x'  y'     ) 

In  order  that  this  equation  reduce  to  the  same  form  as 
that  when  referred  to  its  axes,  it  is  necessary  that  the  term 
containing  x'.  y  disappear.  As  a  and  a'  are  indeterminate, 
we  may  give  to  them  such  values  as  to  reduce  its  co-efficient 
to  zero,  which  gives  the  condition 

A2  sin  a  sin  a'  +  B2  cos  a  cos  a'  =  o, 
and  the  equation  of  the  ellipse  becomes 

(A2  sin  V  f  B2  cos  V)  y'2  +  (A2  sin  2a  -f  B2  cos  8a) 
x'2  =  A1  B2. 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  1«27 

178.  The  condition  which  exists  between  a  and  a  is  not 
sufficient  to  determine  both  of  these  angles.     It  makes  known 
one  of  them,  when  the  other  is  given.     We  may  then  assume 
one  at  pleasure,  and   consequently  there  exists  an   infinite 
number  of  conjugate  diameters. 

179.  The  axes  of  the  ellipse  enjoy  the  property  of  being 
conjugate  diameters,  for  the  relation    between  a  and  a'  is 
satisfied  when  we  suppose  sin  a  —  o,  and  cos  a  =  o,  which 
makes  the  axis  of  x'  coincide  with  that  of  x,  and  y'  with  that 
of  y.     These  suppositions  reduce  the  equation  to  the  same 
form  as  that  found  for  the  ellipse  referred  to  its  axes.     Or, 
these  conditions  may  be  satisfied  by  making  sin  a!  =  o,  and 
cos  a  =  o,  which  will  produce  the  same  result,  only  x  will 
become  y,  and  ?/',  x. 

180.  The  axes  are  the  only  systems  of  conjugate  diameters 
at  right  angles  to  each  other.     For,  if  wre  have  others,  they 
must  satisfy  the  condition 

a'  —  a  =  90°,   or  a'  =  90°  +  a, 
which  gives 

sin  a'  =  sin  90°  cos  a  -f  cos  90°  sin  a  =  +  cos  a, 
cos  a'  =  cos  90°  cos  a  —  sin  90°  sin  a  =  —  sin  a ; 

but  these  values  being  substituted  in  the  equation  of  condition 

A2  sin  a  sin  a'  +  B2  cos  a  cos  a'  =  o, 
it  becomes 

(A2  —  B2)  sin  a  cos  a  =  o, 

which  can  only  be  satisfied  for  the  ellipse  by  making  sin  a  =  o, 
or  cos  a  =  o,  suppositions  which  reduce  to  the  two  cases  just 
considered. 


128  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

181.  If  we  make  A2  —  B2  =  o,  we  shall  have  A  =  B,  the 
ellipse  will  become  a  circle,  and  the  equation  of  condition 
being  satisfied,  whatever  be  the  angle  «,  it  follows  that  all  the 
conjugate  diameters  of  the  circle  are  perpendicular  to  each 
other. 

182.  Making,  successively,  x'  =  o,   and  y'  =  o,  we  shall 
have  the  points  in  which  the  curve  cuts  the  diameters  to 
which  it  is  referred.     Calling  these  distances  A'  and  B',  we 
find 


-- 

~     2  1 


A2  sin  2«  +  B2  cos  2«  ~  A2  sin  V  +  B1  cos  V 

and  the  equation  of  the  ellipse  becomes 

A'V2  +  B'V2  =  A/2B'2, 
2A'  and  2B'  representing  the  two  conjugate  diameters. 

183.  The  parameter  of  a  diameter  is  the  third  propor- 

2B'2 

tional  to  this  diameter  and  its  conjugate;  —  7-7-  is  therefore 

A 

2A'2 
the  parameter  of  the  diameter  2A',  and  -^7-  is  that  of  its 

conjugate  2B'. 

184.  If  we  multiply  the  values  of  A'2  and  B  2  (Art.  182) 

together,  we  get 

A4B4 

A  «iya_  __  _          __  , 

~~  A4  sinV  sin2a  +  A2  Ba  (sin2a  cos2a'  +  cos  2a  sinV)  +  B4  cos'acos^a 

By  adding  and    subtracting   in   the   denominator  of  the 
second  member  the  expression 

2A2  B2  sin  a  sin  a'  cos  a  cos  a', 
and  observing  that 

8m*  (a'  —  a)  =  sin  sa  cos  V  —  2  sin  a  sin  a'  cos  a  cos  a'  + 
sin  V  cos  2a, 


CHAP.  IV.] 
we  have 


ANALYTICAL  GEOMETRY. 


129 


(AB  sin  a'  sin  a  -f  B8  cos  a'  cos  a)2  +  A2  B2  sin2  (a' —  a) 
But  we  have,  from  Art.  180, 

A*  sin  a'  sin  a  +  B2  cos  a'  cos  a  =  o, 
and  reducing  the  other  terms  of  the  fraction,  we  have 

A2B2 


A'2B'2  - 


sin2  (a'  —  a) 


which  gives 


AB  =  A'B'  sin  (a'  —  a). 


(a'  —  a)  is   the   expression  of  the   angle   B'AC'  which    the 

two  conjugate   diameters 

make    with    each    other 

A'B'  sin  (a'  —  a)  expresses 

therefore  the  area  of  the 

parallelogram    Ac'    R'B', 

since  K  sin  (a' —  a)  is  the 

value  of  the  altitude  of 

this  parallelogram.     This 

area  being  equal  to  the  rectangle  AC  RB  formed  on  the  axes, 

we  conclude,  that  in  the  ellipse,  the  parallelogram  constructed 

on  any  two  conjugate  diameters  is  equivalent  to  the  rectangle 

on  the  axes. 

185.  The  equation  of  condition  between  the  angles  a  and 
of  being  divided  by  cos  a  cos  a',  becomes 

A2  tang  a  tang  a  +  B2  =  o.      (1) 

We  may  easily  eliminate  by  means  of  this  equation  the 
angle  a  from  the  value  of  B'2,  or  the  angle  a  from  A".     For 


130  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

this  purpose  we  have  only  to  introduce  the  tangents  of  the 
angles  instead  of  their  sines  and  cosines.  Since  we  have 
always 

.  tang2a  1 

sin  a  =  •= — 5-  ;  cos  2a  =  -= =-  ; 

1  +  tang2a  1  +  tangV 

tangV  1 

sin  a  =  -j— - — - — g-> ;  cos  a  =  •= =-;  • 

1  +  tang  a  1  -{-  tang  V 

Substituting  these  values  in  the  expressions  for  A'2  and  B'a 
lArt.  182),  we  have 

A2- 


tang_'a)  .  A2B2(l  +  tangV) 

a  +  B2  A2  tan   V  +  Ba 


A2  tang  2a  +  B2  A2  tang 

To  eliminate  a  we  have  only  to  substitute  for  tang  a  its 

B2 

value  deduced  from  equation  (1),  tang  a'  =  —  -r-^  --  ,  arid 

A   tang  a 

after  reduction,  the  value  of  B'2  becomes 

_   A4  tang  2a  +  B4 
~  A*  tang  2a  +  B2  ' 

Adding  this  equation  to  the  value  of  A'2,  the  common  nu 
merator 

A2  B2  +  A2  B2  tang  2a  +  A4  tang  2a  +  B4 

may  be  put  under  the  form 

B2  (A2  +  B2)  +  A2  tang2a  (B2  +  A2), 
or  (A2  +  B2)  (A2  tang  2a  +  B2), 

and  the  same  after  reduction  becomes 


that  is,  in  the  ellipse  the  sum  of  the  squares  of  any  two  con* 
jugate  diameters  is  always  equal  to  the  sum  of  the  squares  of 
the  two  axes. 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  131 

186.  The  three  equations 

A2  tang  a  tang  a'  +  B2  =  o, 
AB  =  A'B'sin(a'  — a), 

A2  +  B2  =  A'2  +  B2, 

suffice  to  determine  three  of  the  quantities  A,  B,  A',  B',  a,  a', 
when  the  other  three  are  known.  They  may  consequently 
serve  to  resolve  every  problem  relative  to  conjugate  diam 
eters,  when  we  know  the  axes,  and  reciprocally. 

187.  Comparing  the  first  of  these  equations  with  the  rela 
tions  found  in  Art.  160 ;  when  two  lines  are  drawn  from  the 
extremities  of  the  transverse  axis  to  a  point  of  the  ellipse, 
we  see  that  the  angles  a,  a',  satisfy  this  condition,  since  in 

B2 

both  cases  we  have  aa'  =  —  -r,  •  It  is  then  always  pos 
sible  to  draw  two  supplementary  chords  from  the  vertices  of 
the  transverse  axis,  \vhich  shall  be  parallel  to  two  conjugate 
diameters. 

188.  From  this  results  a  simple  method  of  finding  two 
conjugate  diameters,  which  shall  make  a  given  angle  with 
each  other,  when  we  know  the  axes.     On  one  of  the  axes 
describe  a  segment  of  a  circle  capable  of  containing  the  given 
angle.     Through  one  of  the  points  in  which  it  cuts  the  ellipse 
draw  supplementary  chords  to  this  axis.     They  will  be  par 
allel  to  the  diameters  sought,  and  drawing  parallels  through 
the  centre  of  the  ellipse,  we  shall  have  these  diameters.    The 
construction  should  be  made  upon  the  transverse  axis,  if  the 
angle  be  obtuse;  and  on  the  conjugate,  if  it  be  acute.    When 
the  angle  exceeds  the  limit  assigned  for  conjugate  diameters, 
the  problem  becomes  impossible. 


132 


ANALYTICAL  GEOMETRY. 


[CHAP.  IV 


189.  To  apply  this  principle,  let  it  be  required  to  construe 
two  conjugate  diameters  making  an  angle  of  45°  with  each 
other. 

Upon  the  congregate  axis  BB' 
construct  the  segment  BMM'B' 
capable  of  containing  the  given 
angle.  This  is  done  by  draw 
ing  B'E,  making  EB'G  =  45°. 
BO  perpendicular  to  BE  will 
give  O,  the  centre  of  the  required 
segment,  the  radius  of  which 
will  be  B'O ;  for  the  angle  BMB' 
being  measured  by  half  of  BAB' 
=  45.°  Hence  BM  and  B'M  will  be  supplementary  chords, 
making  with  each  other  the  required  angle ;  and  the  diam 
eters  CF,  CF',  parallel  to  these  chords,  will  be  the  conjugate 
diameters  required  (Art.  168). 


Of  the  Polar  Equation  of  the  Ellipse,  and  of  the  measure 
of  its  surface. 

190.  To  find  the  polar  equation  of  the  Ellipse,  let  o  be 
taken  as  the  pole,  the  co-ordinates  of 
which  are  a  and  b.  Taking  OX'  parallel 
to  CA'  the  formulas  for  transformation 
are  (Art,  122). 

x  =  a  +  r  cos  v,     y  =  b  +  r  sin  v. 

Substituting  these  values  of  x  and  y, 
in  the  equation  of  the  ellipse, 

A2*/2  +  BV  =  A2B2, 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  133 

it  becomes 


A2  sin2*; 
+  B2  cos  *v 


r2  +  2A26sin 
+  2B2a  cos  v 


2a2  —  A2B2  -  o, 


which  is  the  polar  equation  of  the  ellipse. 

191.  If  the  pole  be  taken  at  the  centre  of  the  ellipse,  we 

shall  have 

a  =  o,  and  b  =  o ; 

and  the  equation  becomes 

(A2  sin  2u  +  B2  cos  2v)  r2  =  A2B2. 

192.  If  the  pole  be  taken  on  the  curve,  this  condition 
would  require  that 

A262  +  B2a2  —  A2B2  =  o, 
and  the  polar  equation  would  reduce  to 
(A2  sin  *v  +  B2  cos  2u)  i*  +  (2A26  sin  v  +  2B2a  cos  v)  r  =  o. 

The  results  in  this  and  the  last  article  may  be  discussed  in 
the  same  manner  as  in  the  polar  equation  of  the  circle. 

193.  Let  us  now  suppose  the  pole  to  be  at  one  of  the  foci, 
the   co-ordinates   of  which   are   b  =  o,  a  =  4-  \/A2  —  B2. 
These  values  being  substituted  in  the  general  polar  equation, 
it  becomes 

(A2  sin  2u  +  B2  cos  2r)  r2  +  2B20  cos  v.  r  =  B4. 

Resolving  this  equation  with  respect  to  r,  the  numerate,   of 
the  quantity  under  the  radical  becomes 

B4  (A2  sin  2u  +  B2  cos  2v)  +  B4  a2  cos  2u  ; 
and  putting  for  a2  its  value  A2  —  B2,  it  reduces  to 

A2B4  (sin  2u  +  cos  2u),  or  A2B4 : 
12 


134  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

and  we  have  for  the  two  values  of  r, 

B2  (a  cos  v  —  A) 


B2  cos  2u 


B2  (a  cos  v  +  A) 
~A2sm^  +  B'cosV 

which  may  be  put  under  another  form,  for  we  have 
A2  sin  *«  +  W  cos  *v  =  A2  —  (A2  —  B2)  cos  2u  =  A2  —  «2  Cos»» 

=  (A  —  «cos  i;)  (A  +  a  cos  ?;). 
Making  the  substitutions,  and  reducing,  we  have 
B2  B2 


A  +  a  cos  v  "A  —  a  cos  v 

194.  If  now  the  pole  be  at  the  focus  F,  for  which  a  is 
positive  and  less  than  A,  as  the  cos  v  is  less  than  unity,  the 
product  a  cos  v  will  be  positive  and  less  than  A,  so  that 
whatever  sign   cos  v  undergoes  in  the  different  quadrants, 
A  +  a  cos  v,  and  A  —  a  cos  v,  will  be  both  positive.      The 
first  value  of  r  will  then  be  always  positive  and  give  real 
points  of  the  curve,  while  the  second  will  be  always  negative, 
and  must  be  rejected  (Art.  124).     The  same  thing  takes  place 
at  the  focus  F',  for  although  a  is  negative  in  this  case,  a  cos  v 
will  be  always  less  than  A,  and  the  denominators  of  the  two 
values  will  be  positive.     The  first  value  alone  will  give  real 
points  of  the  curve. 

195.  If,  for  more  simplicity,  we  make 

A2  — B2_ 

""A2"    ~e*' 

we  shall  have 

B2=A2(1— e2), 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  135 

These  values  being  substituted  in  the  positive  value  of  r, 
give 


A  (1-6*) 

1  +  e  cos  v 


T  = 


1  —  e  cos  v ' 


These  formulas  are  of  frequent  use  in  Astronomy. 

196.  In  the  preceding  discussion  we  have  deduced  from 
the  equation  of  the  ellipse,  all  of  its  properties ;  reciprocally 
one  of  its  properties  being  known  we  may  find  its  equation. 

For  example,  let  it  be  required  to  find  the  curve,  the  sum 
of  the  distances  of  each  of  its  points  to  two  given  points 
being  constant  and  equal  to  2A. 

Let  F,  F',  be  the  two  given 
points,  and  A  the  middle  of  the 
line  FF'  the  origin  of  co-ordi 
nates.  Represent  FF'  by  2c. 
Suppose  M  to  be  a  point  of  the 
curve,  for  which  AP  =  x,  PM 
—  y,  and  designate  the  dis 
tances  FM,  F'M,  by  r,  r.  We  shall  have 

r2  =  y*  +  (c  — a:)2 ;  r'2  =  f  +  (c  +  x)* 
r  +  r'  =  2A. 

Adding  the  two  first  equations  together,  and  then  subtract 
ing  the  same  equations,  we  shall  have 

r2  +  r'2  =  2  (if  +  x*  +  c2),  r'2  —  ?-2  =  4cx. 
The  second  equation  may  be  put  under  the  form 

(r — r)  (r'  +  r)  =  4cx. 
Substituting  for  r'  +  r  its  value  2A,  we  get 


r'  —  r  = 


136  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

from  which  we  deduce 

,   ex  c.v 

r1  =  A  +  -j-  ,        r  =  A  —  -r-  • 
A  A 

Putting  these  values  in  the  equation  whose  first  member  is 
•"*  4-  r2,  we  have 


or  A2  (jf  +  or2)  —  cV  =  A2  (A2  —  c*). 

When  we  make  x  =  o,  this  equation  gives 
f  =  A?  —  c2, 

which  is  the  square  of  the  ordinate  at  the  origin.  As  c  is 
necessarily  less  than  A,  this  ordinate  is  real,  and  representing 
it  by  B,  we  have 

B2  =  A2  —  c3. 

If  we  find  the  value  of  c  from  this  result,  and  substitute  it 
in  the  equation  of  the  curve,  we  have 

Ay  +  B2*2  =  A2B2, 

which  is  the  equation  of  the  ellipse  referred  to  its  centre  and 
axes. 

197.  We  may  readily  find  the  expression  for  the  area  of 
the  ellipse.  For  we  have  seen  (Art.  157)  that  if  a  circle  be 
described  on  the  transverse  axis  as  a  diameter,  the  relation 
between  the  ordinates  of  the  circle  and  ellipse  will  be 

^_B 
Y~~  A' 

The  areas  of  the  ellipse  and  circle  are  to  each  other  in  the 
same  ratio  of  B  to  A. 


CHAP.  IV.] 


ANALYTICAL  GEOMETRY. 


137 


To  prove  this,  inscribe  in  the 
circumference  BMM'B'any  poly 
gon,  and  from  each  of  its  angles 
draw  perpendiculars  to  the  axis 
BB'.  Joining  the  points  in  which 
the  perpendiculars  cut  the  el 
lipse,  an  interior  polygon  will  be 
formed.  Now  the  area  of  the 
trapezoid  P'N'NP  is 


(</+ 


The  trapezoid  P'M'MP  in  the  circle  has  for  a  measure 


hence, 


P'NNP  :  P'M'MP  :  :  y  :  Y  :  :  B  :  A. 


These  trapezoids  will  then  be  to  each  other  in  the  constant 
ratio  of  B  to  A.  The  surfaces  of  the  inscribed  polygons  will 
also  be  in  the  same  ratio,  and  as  this  takes  place,  whatever 
be  the  number  of  sides  of  the  polygons,  this  ratio  will  be  that 
of  their  limits.  Designating  the  areas  of  the  ellipse  and 
circle  by  s  and  S,  we  will  have 


S~  A 

that  is,  the  area  of  the  ellipse  is  to  that  of  the  circle  as  the 
semi-conjugate  axis  is  to  the  semi-transverse.  Designating 
by  «?:  the  semi-circumference  of  the  circle  whose  radius  is 
unity,  <rr  A2  will  be  the  area  of  the  circle  described  upon  the 
transverse  axis.  \Ve  shall  then  have  for  the  area  of  the  ellipse 

s  =  «.  AB. 
12*  s 


138  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

The  areas  of  any  two  ellipses  are  therefore  to  each  other  as 
the  rectangles  constructed  upon  their  axc$. 


Of  the  Parabola. 

198.  We  have  found  for  the  general  equation  of  intersec 
tion  of  the  cone  and  plane,  referred  to  the  vertex  of  the  cone 
(Art.  150), 

t/2  sin  2v  +  #2  sin  (v  +  u)  sin  (v  —  u)  —  2c#  sin  v  cos  v  cos  u  —  o. 

This  equation  represents  a  parabola  (Art.  130)  when  u  =  v, 
which  gives 

2c#  cos  2y 

V2  sin  2v  —  %cx  sm  v  cos  2v  =  o,  or  if : =  o  ; 

y  sm  v 

for  the  general  equation  of  the  parabola  referred  to  its  vertex. 
Making  y  =  o  to  find  the  points  in  which  it  cuts  the  axis 
of  x,  we  have 

x  =  o, 

hence  the  curve  cuts  this  axis  at  the  origin. 

Making  x  =  ot  determines  the  points  in  which  it  cuts  the 
axis  of  y.     This  supposition  gives 

y*  =  o, 
hence  the  axis  of  y  is  tangent  to  the  curve  at  the  origin. 

199.  Resolving  the  equation  with  respect  to  y,  wre  have 


y  =  ±  cos 
y 


/2c# 

v  \  /  -  -- 
V  smu 


These  two  values  being  equal  and  with  contrary  signs,  the 
curve  is  symmetrical  with  respect  to  the  axis  of  x.  If  we 
suppose  x  negative,  the  values  of  y  become  imaginary,  since 
the  curve  does  not  extend  in  the  direction  of  the  negative 


CHAP.  IV.] 


ANALYTICAL  GEOMETRY. 


139 


abscissas.     For  every  positive  value  of  x,  those  of  y  will  be 
real,  hence  the  curve  extends  indefinitely  in  this  direction. 

200.  The  ratio  between  the  square  of  the  ordinate  r/2  to 
the  abscissa  x,  being  the  same  for  every  point  of  the  curve, 
we  conclude,  that  in  the  parabola  the  squares  of  the  ordinatei 
are  to  each  other  as  the  corresponding  abscissas. 

201.  The   line  AX  is  called   the  axis 
of  the  parabola,  the  point  A  the  vertext 

2c  cos  ~v 

and  the  constant  quantity  — = the 

J      sin  v 

parameter.        For     abbreviation     make 

2c  cos  2u 

— : =  2p,  the  equation  of  the  pa- 
sin  v 

rabola  becomes 


202.  To  describe  the  pa 
rabola,  lay  off  on  the  axis 
AX  in  the  direction  AB,  a  / 

distance  AB  =  2p.  From  — L 
any  point  C  taken  on  the 
same  axis,  and  with  a  radius 
equal  to  CB,  describe  a  cir 
cumference  of  a  circle ;  from  the  extremity  of  its  diameter 
at  P,  erect  the  perpendicular  PM;  and  drawing  through  the 
point  Q,  QM  parallel  to  the  axis  of  x,  the  point  M  will  be  a 
point  of  the  parabola.  For  by  this  construction  we  have 


hence, 


PM  -  AQ,  and  AQ2  =  AB.  AP; 


MP  =  2p.  AP, 


140  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

203.  If  we  take  on  the  axis  of  the  pa 
rabola  the  point  F  at  a  distance  from  the 

vertex  equal  to  -^>  we    shall    have    for 

every  point  M  of  the  parabola,  the  re 
lation 

FM2  =  if- 


hence, 


that  is,  the  distance  of  any  point  of  the  parabola  from  this 
point  is  equal  to  its  abscissa,  augmented  by  the  distance  of 
the  fixed  point  from  the  vertex.  The  point  F  is  called  the 
focus  of  the  parabola.  Hence  we  see  that  in  the  parabola, 
as  well  as  the  ellipse,  the  distance  of  any  of  its  points  from 
the  focus  is  expressed  in  rational  functions  of  the  abscissa. 
It  follows,  from  the  above  demonstration,  that  all  the  points 
of  the  parabola  are  equally  distant  from  the  focus  and  a  line 

BL  drawn  parallel  to  the  axis  of  y,  and  at  a  distance  —from 

the  vertex.  The  line  BL  is  called  the  Directrix  of  the 
Parabola. 

204.  From  this  property  results  a  second  method  of  de* 
scribing  the  parabola  when  we  know  its  parameter.  From 
the  point  A,  lay  off  on  both  sides  of  the  axis  of  y,  distances 
AB  and  AF,  equal  to  a  fourth  of  the  parameter.  Through 
any  point  P  of  the  axis  erect  the  perpendicular  PM,  and 
from  F  as  a  centre  with  a  radius  equal  to  PB,  describe  an 


CHAP.  IV.] 


ANALYTICAL  GEOMETRY. 


141 


arc  of  a  circle,  cutting  PM  in  the  two  points  MM',  these 
points  will  be  on  the  parabola.  For,  from  the  construction, 
we  have 

FM  =  AP  +  AB  =  x  •+  £. 

205.  The  same  property  enables  us  to  describe  the  para 
bola   mechanically.     For   this    purpose,  apply  the    triangle 
EQR  to   the   directrix   BL.      Take 

a  thread  whose  length  is  equal  to 
QE,  and  fix  one  of  its  extremities 
at  E,  and  the  other  at  the  focus  F. 
Press  the  thread  by  means  of  a 
pencil  along  the  line  QE,  at  the 
same  time  slipping  the  triangle  EQR 
along  the  directrix,  the  pencil  will 
describe  a  parabola.  For, 

FM  +  ME  =  QM  +  ME,  or  QM  =  MF. 

206.  If  we  make  x  =  \p  in  the  equation  of  the  parabola, 
we  get 

y*  =  p*,  or  y  =  p,  or  2*/  =  2p. 

Hence  in  the  parabola,  the  double  ordinate  passing  through 
the  focus,  is  equal  to  the  parameter. 

207.  Let  it  be  required  to  find  the  equation  of  a  tangent 
line  to  the  parabola  whose  equation  is 

Let  x"  y"  be  the  co-ordinates  of  the  point  of  tangency, 
they  must  satisfy  the  equation  of  the  parabola,  and  we  have 

y"2  =  Spa;". 


142  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

The  equation  of  the  tangent  line  will  be  of  the  form 

y  —  y"  =  a(x  —  x"). 

It  is  required  to  determine  a. 

Let  the  tangent  be  regarded  as  a  secant,  whose  points  of 
intersection  coincide.  To  determine  the  points  of  intersec 
tion,  the  three  preceding  equations  must  subsist  at  the  same 
time.  Subtracting  the  second  from  the  first,  we  have 

(y-y")(y  +  y")  =  IP  (*—«")• 

Putting  for  y  its  value  drawn  from  the  equation  of  the 
tangent,  we  get 

(2ay"  +  a2  (x  —  x")  —  2p)  (x  —  x")  =  o. 

This  equation  may  be  satisfied  by  making  x  —  x"  =  o, 
which  gives  x  =  x"  and  y  =  y"  for  the  co-ordinates  of  the 
first  point  of  intersection,  or  by  making 

a2  (x  —  x")  —  2p  =  o. 


This  equation  will  make  known  the  other  value  of  x  when 
a  is  known.  But  when  the  secant  becomes  a  tangent,  the 
points  of  intersection  unite,  and  we  have  for  this  point  also 
x  =  x",  which  reduces  the  last  equation  to 


hence, 


- 


Substituting  this  value  in  the  equation  of  the  tangent,  it 
becomes 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  143 

or  reducing  and  observing  that  y"2  =  2px",  \\e  have 


for  the  equation  of  the  tangent  line. 

208.  By  the  aid  of  these  formulas  we  may  draw  a  tangent 
to  the  parabola  from  a  point  without,  whose  co-ordinates  are 
x',  y'.     For  this  point  being  on  the  tangent,  we  have 

y'y"  =p(x'  +  x"), 

and  joining  with  this  the  relation 

y"2=2px", 

we  may  from  these  equations  determine  the  co-ordinates  of 
the  point  of  tangency.  The  resulting  equation  being  of  the 
second  degree,  there  may  in  general  be  two  tangents  drawrn 
to  the  parabola,  from  a  point  without. 

209.  To  find  the  point  in  which 
the  tangent  meets  the  axis  of  x, 
make  y  =  o  in  the  equation 


yy"  = 


we  get 


x  = 


x 


which  is  the  value  of  AT.  Adding  to  it  the  abscissa  AP, 
without  regarding  the  sign,  we  shall  have  the  subtangent, 

PT  =  2*", 

that  is,  in  the  parabola,  the  subtangent  is  double  the  abscissa. 
This  furnishes  a  very  simple  method  of  drawing  a  tangent 
to  the  parabola,  when  we  know  the  abscissa  of  the  point  of 
tangency. 


344  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

210.  The  equation  of  a  line  passing  through  the  point  of 
\angency  is  of  the  form 

In  order  that  this  line  be  perpendicular  to  the  tangent,  for 
which  we  have  (Art.  207), 

~  !?' 
it  is  necessary  that  we  have 

aa'  +  1  =  o, 
hence 

,._£ 

The  equation  of  the  normal  becomes 


Making  y  =  o,  we  have 

x  —  x"  =  p, 

which  shows  that  in  the  parabola  the  subnormal  is  constant 
and  equal  to  half  the  parameter. 

211.  The  directions  of  the  tangent  and  normal  have  re 
markable  relations  with  those  of  the  lines  drawn  from  the 
focus  to  the  point  of  tangency. 

The  equation  of  a  line  passing  through  the  point  of  tan 
gency  is 

y  —  y»  =  a'(x  —  x"), 

and  the  condition  of  its  passing  through  the  focus,  for  which 


CHAP.  IV.]  ANALYTICAL  GEOMETRY. 

-y" 


P      • 
=  o,  x  =  -~  gives 


—  a?' 


The  angle  FMT  which  this  line  makes  with  the  tangent 
has  for  a  trigonometrical  tangent  (Art.  64), 


Substituting  for  a  its  value  — »   and    for    a'  that   foand 
above,  and  observing  that  y"2  =  2px",  we  have 

tang  FMT  =  £  =  a; 

hence,  in  the  parabola,  the  tangent  line  makes  equal  angles 
with  the  axis,  and  with  a  line  drawn  from  the  focus  to  the 
point  of  tangency,  so  that  the  triangle  FMT  is  always 
isosceles ;  consequently,  when  the  point  of  tangency  M  is 
given,  to  draw  a  tangent,  we  have  only  to  lay  off  from  F 
towards  T  a  distance  FT  =  FM.  FM  will  be  the  tangent 
required. 

212.  If  through  M  we  draw  MF'  parallel  to  the  axis,  the 
tangent  will  make  the  same  angle  with  this  line  as  with  the 
axis,  hence  in  the  parabola  the  lines  drawn  from  the  point  of 
tangency  to  the  focus  and  parallel  to  the  axis  make  equal 
angles  with  the  tangent.  From  this  results  a  very  simple 
method  of  drawing  a  tangent  to  the  parabola  from  a  point 
without.  Let  G  be  the  point,  F  the  focus,  BL  the  directrix. 
From  G  as  a  centre,  with  a  radius  equal  to  GF,  describe  a 
circumference  of  a  circle,  cutting  BL,  in  L,  L'.  From  these 
13  T 


146 


ANALYTICAL  GEOMETRY 


[CHAP.  IV. 


points  draw  LM,  L'M',  parallel  to 
the  axis.  M  and  M'  will  be  the 
points  of  tangency,  and  GM,  GM', 
will  be  the  two  tangents  that  may 
be  drawn  from  the  point  G.  For, 
by  the  nature  of  the  parabola  ML 
=  MF,  arid  by  construction  GF 
=  GL,  the  line  GM  has  all  of  its  points  equally  distant  from 
F  and  L.  It  is  therefore  perpendicular  to  the  line  FL,  con 
sequently  the  angle  LMG,  or  its  opposite  ZltfF',  is  equal  to 
the  angle  GMF.  MG  is  therefore  a  tangent  at  the  point  M. 
The  same  may  be  proved  with  regard  to  GM'. 


Of  the  Parabola  referred  to  its  Diameters. 

213.  Let  us  now  examine  if  there  are  any  systems  of 
oblique  co-ordinates,  relatively  to  which  the  equation  of  the 
parabola  will  retain  the  same  form  as  when  it  is  referred  to 
its  axis.  The  general  formulas  for  transformation  are 

x  =  a  +  x  cos  a  +  y'  cos  a',     y  =  b  +  x'  sin  a  +  y  sin  a . 

These  values    being   substituted    in   the   equation  of  the 

parabola 

y2  =  2px, 
it  becomes 

y'2  sin  V  +  2x'y'  sin  a  sin  a'  +  x'9  sin  2a  +  62  —  %ap  ")  _ 
+  2  (b  sin  a'  — p  cos  a)  y  +  2  (b  sin  a  — p  cos  a)  x'  5 

In  order  that  this  equation  preserve  the  same  form  as  the 
preceding,  we  must  have 
sin  a'  sin  a  =  0,  sin  2a  =  o  b  sin  a'  — p  cos  a'  =  o,  b3  —  Sap  =  Q, 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  147 

and  the  equation  reduces  to 

/*.-*,<, 

smV 

and  putting  for     .    2  ,  >  »',  we  have 

sin  V    * 

?/'2  =  2/z'. 

214.  The  second  of  the  preceding  equations  of  conditionj 
shows  that  sin  a.  =  o,  that  is,  the  axis  of  x'  is  parallel  to  the 
axis  of  x.      Hence,  all  the  diameters  of  the  parabola  are 
parallel  to  the  axis. 

215.  The  two  other  equations  give 

b2  =  Zap, 
and 

sin «'  p 

;  =  lang  a   =  -=•  • 

cos  a  o 

The  first  shows  that  the  co-ordinates  a  and  b  of  the  new 
origin  satisfy  the  equation  of  the  parabola.  This  origin  is 
therefore  a  point  of  the  curve.  The  second  determines  the 
inclination  of  the  axis  of  y'  to  the  axis  of  x,  and  shows  that 
this  axis  is  tangent  to  the  curve  at  the  origin,  since  it  makes 
the  same  angle  with  the  axis  of  x  as  the  tangent  line  at  this 

point  (Art.  207),  for  which   a  =  4,  • 

216.  The  equation  y'*  =  Vp'x',  giving  two  equal  values  for 
y',  and  with  contrary  signs  for  each  value  of  #',  each  diameter 
bisects  the  corresponding  ordinates. 

217.  The  equation  of  the  parabola  being  of  the  same  form 
when  referred  to  its  diameters  and  axis,  all  of  its  properties 
which  are  independent  of  the  inclination  of  the  co-ordinates 
will  be  the  same  in  these  two  systems.     Thus,  to  describe  a 


148  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

parabola  when  we  know  the  parameter  of  one  of  its  diameters, 
and  the  inclination  of  the  corresponding  ordinates,  describe 
a  parabola  on  this  diameter  as  an  axis  with  the  given  para 
meter,  and  then  incline  the  ordinates  without  changing  their 
lengths,  we  shall  have  the  parabola  required. 


Of  the  Polar  Equation  of  the  Parabola,  and  of  the 
Measure  of  its  Surface. 

218.  Let  us  resume  the  equation  of  the  parabola  referred 
Lo  its  axis, 


and  take  0  for  the  position  of  the  pole, 
the  co-ordinates  of  which  are  a  and  b  ; 

,_.       draw  OX'  parallel  to  the  axis.     The  for- 

mulas  for  transformation  are  (Art.  122). 


A\      £ 

x  =  a  +  r  cos  v,    y  =  b  +  r  sin  v. 

Substituting  these  values  in  the  equation  of  the  parabola,  it 
becomes 

r*sin2i;  +  2  (b  sin  v—  pcos  v)  r  +  b*  —  2pa  =  c. 

If  the  pole  be  on  the  curve, 

und  the  equation  reduces  to 

r2  sin  zv  +  2  (b  sin  v  — p  cos  v)  r  =  o, 
wh\ch  may  be  satisfied  by  making 

r  =?  o,  or  r  sin  *v  +  2  (b  sin  v  — p  cos  v)  =  o. 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  149 

The  last  equation  gives 

2  (p  cos  v  —  b  sin  v) 


r  = 


If  this  second  value  of  r  were  zero,  the  radius  vector 
would  be  tangent  to  the  curve.  But  this  supposition  requires 
that  we  have 

%p  cos  v  —  25  sin  v  =  o, 
which  gives 

sin  v  p 

=  tang  v  =  Y-» 

cos  v  o 

which  is  the  same  value  found  for  the  inclination  of  the  tan 
gent  to  the  axis  (Art.  207). 

219.  If  the  pole  be  placed  at  the  focus  of  the  parabola, 
the  co-ordinates  of  which  are  b  =  o  a  —  -~  >  the  general 
polar  equation  becomes 

r2  sin  *u  —  2p  cos  v.  r  =  p2 
and  the  values  of  r  are 

p  (cos  v  +  1)  p  (cos  v  —  1) 

T  ==  .      o  '  ^*  —  :      5 

sin  v  „     sin  v 

The  second  value  of  r  being  always  negative,  since  cos 
v  <  1  and  (cos  v  —  1)  consequently  negative,  must  be  re 
jected.  The  first  value  is  always  positive,  and  will  give 
real  points  to  the  curve.  It  may  be  simplified  by  putting 
for  sin  *v,  1  — cos  2v,  which  is  equal  to  (1  -f  cos  v)  (1  —  cos  u), 
and  this,  value  reduces  to 

p  (1  +  cos  v)  __          p 

~  (1  +  cos  v)  (1  —  cos  v)  ~'    I  —  cos  v  ' 
13* 


150  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

which  is  the  polar  equation  of  the  parabola  when  the  pole  is 
at  the  focus. 

220.  If  v  =  o,  r  =  —  =  infinity.     Every  other  value  of  v 

from   zero   to   360°   will   give   finite   values    to   r.      When 
v  =  90°,  cos  v  =  o  and  r  =  p.    When  v  =  180°,  cos  v  =  —  1 

and  r  =  4r »  results  which   correspond  with   those   already 
found. 

221.  In  the  preceding  discussion  we  have  deduced  all  the 
properties  of  the   parabola  from  its  equation ;    reciprocally 
we  may  find  its  equation  when   one  of  these  properties  is 
known. 

Let  it  be  required,  for  example,  to  find  a  curve  such  that 
the  distances  of  each  of  its  points  from  a  given  line  and 
point  shall  be  equal.  Let  F  be  the 
given  point,  BL  the  given  line.  Take 
the  line  FB  perpendicular  to  BL  for  the 
axis  of  x,  and  place  the  origin  at  A,  the 
middle  of  BF,  and  make  BF  =  p. 

For  every  point  M  of  the   curve,  we 
shall  have  these  relations 


FM2  =  y*  +    ^  —    j 

But  by  the  given  conditions  we  have 

FM  -  LM  =  BA  +  AP, 
hence 


eliminating  FM  we  have 

y2 
which  is  the  equation  of  the  parabola. 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  151 

222.  To  find  the  area  of  any  portion  of  the  parabola,  let 
APM  be  the  parabolic  segment  n  g  ar. 

whose   area   is   required.      Draw  q' 

MQ    parallel    and    AQ    perpen 
dicular   to  the   axis.      The    area 
of  the  segment  APM  is  two-thirds      — r — ^ 
of  the  rectangle  APQM.  \ 

Inscribe  in  the  parabola  any  rectilinear  polygon  MM'M". 
From  the  vertices  of  this  polygon  draw  parallels  to  the  lines 
AP,  PM,  forming  the  interior  rectangles  PP'joM;  P'PyM", 
and  the  corresponding  exterior  rectangles  QQ'#  M' .  Repre 
senting  the  first  by  P,  P',  P",  and  the  last  p9  p',  p",  we 
shall  have 

Pi  /  i\  .  /  .* 

—  y  \^ "~°~  y* ))    P  *•—  cc  ft/  — —  \i ) 

which  gives 


P     xy  —  y') 

but  the  points  M,  M',  M",  belong  to  the  parabola,  and  we 
have 

ff=2px,    y'2=2px', 
which  give 

«•-  ->=^--  '-g- 

Substituting  these  values,  the  ratio  of  P  to  p  becomes 


_ 
p     y2(y—y')        y1 

The  same  reasoning  will  apply  to  all  of  the  interior  and 
corresponding  exterior  rectangles,  and  we  have  the  equations 


152  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

Py_y'  +  y" 
p  '      y" 

P"       y"  +  y'" 

—  =  -  --  BT-  »  &c- 

p        y 

The  polygon  M,  M',  M",  being  entirely  arbitrary,  we 
may  place  the  vertices  in  such  a  manner  that  designating  by 
u  any  constant  quantity,  we  have  always 

y  —  y'  =  uy' 

y'—y"  =  uy" 

y"  —  y'"  =  uy'",&.c. 

which  is  equivalent  to  making  y,  y',  y",  decrease  in  a  geo 
metrical  progression.  But  from  this  supposition  we  nave,  by 
adding  2?/'  to  the  members  of  the  first  equation,  %y"  to  those 
of  the  second,  &c., 


and  the  several  ratios  become 


~  =  2  +  u, 


P" 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  153 

Hence  these  ratios  will  be  equal,  whatever  be  u.     By  com 
position  we  have 

P  +  P'  +  P"  +  &c. 


but  the  numerator  of  the  first  member  is  the  sum  of  the  in 
terior  rectangles,  and  the  denominator  that  of  the  exterior 
rectangles.  As  u  diminishes,  this  ratio  approaches  more  and 
more  the  value  of  2,  and  we  may  take  u  so  small,  that  the 
difference  will  be  less  than  any  assignable  quantity.  But, 
under  this  supposition,  the  inscribed  and  circumscribed  rec 
tangles  approach  a  coincidence  with  the  inscribed  and  cir 
cumscribed  curvilinear  segments,  consequently  the  limit  of 
their  ratio  is  equal  to  the  ratio  of  the  segments,  and  repre 
senting  the  first  by  S,  and  the  second  by  s,  we  have 


!-«• 


which  gives 


and  dividing  these  equations  member  by  member, 
S  =  I  (S  +  .)  ; 

but  S  +  s  is  the  sum  of  the  inscribed  and  circumscribed 
segments,  and  is  consequently  the  surface  of  the  rectangle 
APMQ.  Hence,  the  area  of  the  parabolic  segment  APM  is 
equal  to  two-thirds  of  the  rectangle  described  upon  its  abscissa 
and  ordinate. 

223.   Quadrable  Curves  are  those  curves  any  portion  of 

u 


154  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

whose  area  may  be  expressed  in  a  finite  number  of  alge 
braic  terms.  The  parabola  is  quadrable,  while  the  ellipse 
is  not. 

Of  the  Hyperbola. 

224.  We  have  found  (Art.  150)  for  the  general  equation 
of  the  conic  sections, 

t/2  sin  *v  +  xz  sin  (v  +  u)  sin  (v  —  u)  —  2c#  sin  v  cos  v  cos  u  =  o, 

and  (Art.  131)  that  this  equation  represents  a  class  of  curves 
called  Hyperbolas,  when  u  >  v. 

To  discuss  this  curve,  let  us  find  the  points  in  which  it 
cuts  the  axis  of  x;  make  y  =  o,  we  have 

x2  sin  (v  +  u)  sin  (v  —  u)  —  2cx  sin  v  cos  v  cos  u  =  o, 
which  gives  for  the  two  values  of  x 

2c  sin  v  cos  v  cos  u 

X  =  09    X  —  — 7 ; r : -. r  9 

sin  (v  +  u)  sin  (v  —  u) 

which  show  that  the  curve 
cuts  this  axis  at  two  points 
B,B',  one  of  which  is  at  the 
origin,  and  the  other  at  a  dis- 

__,       2c  sin  v  cos  v  cos  u 
tance  BB  =  -r- 


sin  (v  +  u)  sin  (i> — u) 
from  the  origin,  and  on  the  negative  side  of  the  axis  of  y, 
since  sin  (v  —  u)  is  negative.  Making  x  =  o,  we  find 

y°  =  o; 

hence  the  axis  of  y  is  tangent  to  the  curve  at  the  origin. 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  155 

225.  Resolving  this  equation  with  respect  to  y,  we  have 

1 

y  —  ~^~v  V —  x2  sin  (v  +  u)  sin  (v  —  u)  +  %cx  sin  v  cos  v  cos  u. 

These  two  values  being  equal,  and  with  contrary  signs, 
the  curve  is  symmetrical  with  respect  to  the  axis  of  x.  For 
every  positive  value  of  x,  we  shall  have  a  real  value  of  y, 
since  sin  (v  —  u)  being  negative,  the  sign  of  the  first  term 
under  the  radical  is  essentially  positive.  The  curve  therefore 
extends  indefinitely  on  the  positive  side  of  the  axis  of  y.  If 
x  be  negative,  y  will  only  have  real  values  when  — or2  sin 
(v  -f  u)  sin  (v  —  u)  >  %cx  sin  v  cos  v  cos  u.  Putting  the  value 
of  y  under  the  form 

y=       

1         /  /         2csinucos  t?cos  w\ 

- \/  X  Sin  (V  +  U)  Sin  (V U)  (X ~r— ; r-^— : )• 

sin  v  V  '  \        Bin(t?+tt)sin(c — */ 

Since  sin  (u  —  u)  is  negative,  the  first  factor 
—  x  sin  (v  +  u)  sin  (v  —  u) 

will  be  negative  for  every  negative  value  of  a?.  The  sign  of 
the  quantity  under  the  radical  will  then  depend  upon  that 
of  the  second  factor 

(2c  sin  v  cos  v  cos  u  \ 
sin  (v  +  u)  sin  (v  —  uy 

But  this  factor  will  be  positive  so  long  as 
2c  sin  v  cos  v  cos  u 

<£     <Q f 

sin  (v  +  u)  sin  (v  —  u) 
since 

2c  sin  v  cos  v  cos  u 


sin  (u  +  u)  sin  (u  —  tt 
is  essentially  positive. 


156  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

But   for   negative   values   of  x   which   are   greater   than 

2c  sin  v  cos  v  cos  u 

-.  —  7  —  :  —  r-  -  —  7  -  r   the   second    factor  will  be  negative, 
sin  (v  +  u)  sin  (v  —  u) 

and  the  quantity  under  the  radical  positive.     The  values  of 
y  will  therefore  be  imaginary  for  negative  values  of  #  between 

the  values 
) 

2c  sin  v  cos  v  cos  u 

X  —  0    and  X  =    -.  -  -—  —  -  r—  :  -  ;  -  -  r  ? 

sin  (v  +  u)  sin  (v  —  u) 

that  is,  between  B'  and  B';  and  real  for  all  negative  values 

2c  sin  v  cos  v  cos  u 

of  x  greater  than  —.  —  -.  —  -  —  r—  ^  —  -.  -  r 
sin  (v  +  u)  sin  (u  —  u) 

There  is  therefore  no  part  of  the  curve  between  B,  and  B', 
but  it  extends  indefinitely  from  B'  negatively. 

226.  Let  the  origin  of  co-ordinates  be  taken  at  A,  the 
middle  of  BB'. 

The  formula  for  transformation  is, 

c  sin  v  cos  v  cos  u 

*  sin  <«  +  «)  dn  (•,--«)  is  essentially  neSatlve' 
c  sin  v  cos  v  cos  u 

ft    s—     ftf     _|_     ____  _____  . 

sin  (  v  -f-  u)  sin  (v  —  u) 

Substituting  this  value  of  x  in  the  equation  of  the  curve, 
and  reducing,  we  have 


sin 


. 
s\n(v  +  u)sm(v  —  u) 


Making  y  =  o,  to  find  the  point  in  which  it  cuts  the  axis  of 
x,  we  find 

c  sin  v  cos  v  cos  u 


X'  —  AB  = 


-r-T  -  ;  -  r^  —  ;  -  - 

sm  (u  +  w)  sin  (v  —  u) 


but  for  x'  —  o,  we  find  that  the  values  of  y  are  imaginary  ; 
the  curve  therefore  does  not  intersect  the  axis  of    . 


CHAP  IV.]  ANALYTICAL  GEOMETRY.  157 

If  we  make 

2  c2  sin  *v  cos  *v  cos  *u 

sin2  (v  +  u)  sin2  (v  —  u) 
_  c2  cos  2u  cos  2& 


sin  (v  +  u)  sin  (v  —  u) 
and  multiply  the  two  members  of  the  equation  (1)  by 

c2  cos2t;  cos*u 
sin2  (v  +  u)  sin2  (v  —  u)  ' 

and  put  x  for  #',  we  shall  have 

A2/  — BV  =  —  A2B2 

for  the  equation  of  the  hyperbola  referred  to  its  centre  and 
axes. 

227.  The  quantities  2A,  2B,  are    called  the  axes  of  the 
hyperbola.     The  point  A  is  the  centre.     Every  line  drawn 
through  the  centre  and  terminated  in  the  curve  is  called  a 
diameter,  and  there  results  from  the  symmetrical  form  of  the 
hyperbola  that  every  diameter  is  bisected  at  the  centre. 

228.  The  equation  of  the  ellipse  referred  to  its  centre  and 
axes,  is 

Ay  +  BV  =  A2B2. 

Comparing  this  equation  with  that  of  the  hyperbola,  we 
see  that  to  pass  from  one  to  the  other  we  have  only  to  change 
B  into  B  V  —  1.  This  simple  analogy  is  important  from  the 
facility  it  affords  in  passing  from  the  properties  of  the  ellipse 
to  those  of  the  hyperbola. 

229.  When  the  two  axes  of  the  hyperbola  are  equal,  its 
equation  becomes 

t/'-^-A'; 

we  say  then  that  the  hyperbola  is  equilateral. 
14 


158 


ANALYTICAL  GEOMETRY. 


[CHAP.  IV. 


When    the    axes    of  the    ellipse    are    equal,  its    equation 
becomes 

which  is  the  equation  of  a  circle.  The  equilateral  hyper 
bola  is  then  to  the  common  hyperbola  what  the  circle  is  to 
the  ellipse. 

230.  It  follows  from  this  analogy  between  the  ellipse  and 
hyperbola,   that   if  these  curves  have  equal   axes   and  are 
placed  one  upon  the  other,  the  ellipse  will  be  comprehended 
within   the   limits,  between   which    the    hyperbola  becomes 
imaginary,  and    reciprocally,  the   hyperbola  will  have  real 
ordinates,  when  those  of  the  ellipse  are  imaginary. 

231.  The  equation  of  a  line  passing  through  the  point  B', 

for  which  y  —  o,  x  —  —  A,  is 

y  =  a  (x  +  A). 

That  of  a  line  passing  through 
B,  for  which  y  =  o,x=  +  A,  is 


In  order  that  these  lines  intersect  on  the  hyperbola,  these 
equations  must  subsist  at  the  same  time  with  that  of  the 
hyperbola.  Multiplying  them  member  by  member,  we  have 
f  =  oaf  (x2  —  A2). 

Combining  this  with  the  equation  of  the  hyperbola,  put 
under  the  form 


we  have 


W 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  159 

which  establishes  a  constant  relation  between  the  tangents  of  the 
angles  which  the  supplementary  chords  make  with  the  axis  of  .r. 

232.  When  the  hyperbola  is  equilateral  B  =  A,  and  this 
relation  reduces  to 

ad  —  1, 
hence 

1 
a=^' 

or  tang  a  =  cot  a', 

which  shows  that  in  the  equilateral  hyperbola,  the  sum  of  the 
two  acute  angles  which  the  supplementary  chords  make  with 
the  transverse  axis,  on  the  same  side,  is  equal  to  a  right  angle. 

233.  If  we  put  x  in  the  place  of  y  and  y  for  x  in  the  equa 
tion  of  the  hyperbola,  it  becomes 

By  —  AV=  A2B2. 

If  in  this  equation  we  make  x  =  o,  y  becomes  real,  and 
y  =  o  makes  x  imaginary.  Hence 
the  curve  cuts  the  axis  of  y,  but  does 
not  meet  with  that  of  x.  It  is  then 
situated  as  in  the  figure,  the  trans 
verse  axis  being  b,  b'.  The  curve  is 
said  to  be  referred  to  its  conjugate 
axis,  because  the  abscissas  are  reck 
oned  on  this  axis. 

231.  The  analogy  between  the  ellipse  and  hyperbola, 
leads  us  to  inquire  if  there  are  not  points  in  the  hyperbola 
corresponding  to  the  foci  of  the  ellipse. 

In  the  ellipse  the  abscissas  of  these  points  were 


x  =  ±  VA2  —  B2. 


160  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 


Changing  B  into  B  V —  1,  we  have  for  the  hyperbola 


Let  us  for  simplicity  make 


and  let  F,  F',  be  the  points  at  this 
distance  from  the  centre,  we  will 
have 


(x  —  cf  =       (a*—  A1)  +  *a  — 

A. 


from  which  we  obtain 


In  the  same  manner  we  will  have 
F'M  =  f  +  A, 

that  is,  the  distances  FM,  F'M,  are   expressed   in   rational 
functions  of  the  abscissa  x. 

Subtracting  these  equations  from  each  other,  we  get 

F'M'  —  FM  =  2A. 

Hence,  the  difference  of  these  distances  is  constant  and 
equal  to  the  transverse  axis. 

235.  To  find  the  position  of  the  foci  geometrically,  erect 
at  one  of  the  extremities  of  the  transverse  axis  a  perpen 
dicular  BE  =  B  the  semi-conjugate  axis,  and  draw  AE. 
From  the  point  A  as  a  centre  with  a  radius  AE,  describe  a 
circumference  of  a  circle,  cutting  the  axis  in  F,  F'.  These 
points  are  the  foci  of  the  hyperbola. 


ANALYTICAL  GEOMETRY. 


161 


CHAP.  IV.] 

236.  The  preceding  properties  enable  us  to  construct  the 
hyperbola,     From  the  focus 

F  as  a  centre  with  a  radius 

BO,  describe  a  circumference 

of  a  circle.     From  F'  as  a 

centre  with  a  radius  BO  = 

BB'  +  BO  describe   another 

circumference.      The  points 

M,  M',  in  which  they  intersect,  are  points  of  the  hyperbola, 

for 

FM  —  FM  =  2A. 

237.  By  following  the  same  course  explained  in  Art.  165, 
for  the  ellipse,  we  may  find  the  equation  of  a  tangent  line  to 
the  hyperbola.     But  this  equation  may  be  at  once  obtained 
by  making  B  =  B  \/ —  1  in  the  equation  of  a  tangent  line  to 
the  ellipse,  and  we  have 

A*yy"  —  Wxx"  =  —  A2B2 

for  the  equation  of  a  tangent  line  to  the  hyperbola. 

238.  The  equation  of  a  line  passing  through  the  centre 
and  point  of  tangency  is 

y"  =  a'*", 

which  gives 

a'  =  ^'-  -! 

Multiplying  this  by  the  value  of  a  corresponding  to  the  tan- 

BV 

gent,  which  is  a  =  -rv-ja »  we  have 
Ay 

aa!  =  -TV 


14* 


162  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

Comparing  this  result  with  Art.  231,  we  find  the  same  value 
for  aa.  Hence,  the  angles  which  the  supplementary  chords 
make  with  the  axis  of  a?,  are  equal  to  those  which  the  tangent 
line  and  the  diameter,  drawn  through  the  point  of  tangency, 
make  with  the  same  axis.  The  supplementary  chords  are 
therefore  parallel  to  the  tangent  line  and  this  diameter. 


Hence,  to  draw  a  tangent  line  to  the  hyperbola  at  any  point 
M,  draw  the  diameter  AM,  then  through  B'  draw  the  chord 
B'N  parallel  to  AM  ;  MT  parallel  to  BN  will  be  the  tangent 
required. 

Of  the  Hyperbola  referred  to  its  Conjugate  Diameters. 

239.  The  properties  of  the  hyperbola  referred  to  its  diam 
eters  may  be  easily  deduced  from  those  of  the  ellipse.  By 
making  B'  =  B'  V  —  1  in  the  equation  of  the  ellipse  (Art. 
182),  we  find 

AV  —  B'V2  =  —  A'2B'2. 

The  quantities  2Af,  2B',  are  called  the  conjugate  diameters 
of  the  hyperbola. 


CHAF.  IV.]  ANALYTICAL  GEOMETRY.  183 

This  equation  could  be  also  obtained  by  the  same  method 
demonstrated  for  finding  the  equation  of  the  ellipse. 

240.  In  the  same  mariner,  by  making  B  =  B  V  —  1,  and 
B'  =  B  V  —  1    in    the    equations    Art.  186,  we   have   the 
relation 

A'2  — B'2  =  A2  — B2, 
A'B'  sin  (a'  —  a)  =  AB, 
A2  tang  a  tang  a'  —  B2  =  o. 

The  first  signifies  that  the  difference  of  the  squares  con 
structed  on  the  conjugate  diameters  is  always  equal  to  the 
difference  of  the  squares  constructed  on  the  axes.  Hence  the 
conjugate  diameters  of  the  hyperbola  are  unequal.  The 
supposition  of  A'  =  B'  gives  A  =  B,  and  reciprocally.  The 
equilateral  hyperbola  is  the  only  one  which  has  equal  conjugate 
diameters. 

The  second  of  the  preceding  equations  shows  that  the 
parallelogram  constructed  on  the  conjugate  diameters  is  al 
ways  equivalent  to  the  rectangle  on  the  axes. 

The  third  relation  compared  with  that  of  Art.  248,  shows 
that  the  supplementary  chords  drawn  to  the  transverse  axis  are 
respectively  parallel  to  two  conjugate  diameters. 

Of  the  Asymptotes  of  the  Hyperbola,  and  of  the  Properties 
of  the  Hyperbola   referred  to  its  Asymptotes. 

241.  The  indefinite  extension  of  the  branches  of  the  hyper 
bola  introduces  a  very  remarkable  law  which  is  peculiar  to 
it.     The  equation  of  the  hyperbola  referred  to  its  centre  and 
axes  may  be  put  under  the  form 


164  ANALYTICAL  GEOMETRY.  [CHAP.  IV 

which  gives  for  the  two  values  of  y, 

Ex 


Developing  the  second  member,  it  becomes 

A2          A4  A6 


Bo? 

and  multiplying  by  ±  -r-  >  it  becomes 
A 

BA         BA5          BAS 


In  proportion  as  a?  augments  A,  and  B  remaining  constant, 

BA    BA3 

the  terms  -  »  —  5-  >  &c.,  will  diminish.     The  values  of  y 

Ex 
will  continually  approach  to  those  of  the  first  term  ±  —  r-  . 

As  a?  is  indefinite,  we  may  give  it  such  a  value  as  to  make 
the  difference  smaller  than  any  assignable  quantity.  If, 
therefore,  we  construct  the  two  lines  whose  equations  are 
represented  by 

Bo?  Ex 


these  lines  will  be  the  limits  of  the  branches  of  the  hyper 
bola,  which  they  will  continually  approach  without  ever 
meeting.  And  this  may  be  readily  shown,  for  we  have 

BV 

y2  =  -A2  —  B2  for  points  on  the  hyperbola; 
A. 

BV 

t/2  =  —  r-g-  for  points  on  the  lines  ; 
A 

which  shows  that  the  ordmates  corresponding  to  the  same 
abscissas  are  always  smaller  for  the  curve  than  for  the  lines. 
These  lines  are  called  Asymptotes. 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  165 

242.  We  can  easily  prove  from  the  preceding  expressions 
that  the  asymptotes  continually  approach  the  hyperbola; 
for,  subtracting  the  first  from  the  second,  and  designating 
the  ordinates  of  the  asymptotes  by  y',  we  have 

^*_y  =  B», 

or, 

(</'-</)  (</  +  2/)  =  B>; 
hence, 


y'  —  y  is  the  difference  of  the  ordinates  of  the  asymptotes 
and  hyperbola.  The  fraction  which  expresses  this  value  has 
a  constant  numerator,  while  the  denominator  varies  with  y 
and  y  '.  The  more  y  and  y  increase,  the  smaller  will  be  this 
difference.  As  there  is  no  limit  to  the  values  of  y  and  y1  , 
the  difference  may  be  made  smaller  than  any  assignable 
quantity. 

243.  To  construct  the  asymptotes  of  the  hyperbola,  draw 
through  the  extremity  of  the  transverse  axis  a  perpendicular, 
on  which  lay  off  above  and  below  the  axis  of  #  two  distances 
equal  to  half  of  the  conjugate  axis.     Through  the  centre  of 
the  hyperbola  and  the  extremities  of  these  distances,  draw 
two  lines  ;    they  will  be  the  asymptotes  required,  for  they 
make  with  the  axis  of  x,  angles  whose  trigonometrical  tan- 

B 
gents  are  d=  -r  . 

244.  If   the  hyperbola   be   equilateral,   B  =  A,    and   the 
asymptotes  make  angles  of  45°  and  135°  with  the  aa.is  of  x. 

245.  The  asymptotes  are  the  limits  of  all  tangents  drawn 
to  the  hyperbola.     In  fact,  the  equation  of  a  tangent  line  To 
this  curve  being  (Art.  237), 


166  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

A*yy"  —  Wxx"  =  —  A2B-, 

the  point  in  which  it  meets  the  axis  of  the  hyperbola,  has  for 

an  abscissa 

_  A^ 
X~x"  ' 

In  proportion  as  x"9  which  is  the  abscissa  of  the  point  of 
tangency,  increases,  the  value  of  x  diminishes  ;  and  when 
x"  —  infinity,  x  =  o.  In  this  supposition  the  value  of  y"  be 

comes  also  infinite  and  equal  to  rb  —  -r—  >    so    that,    substi- 

A 

tuting  this  value  in  the  expression  for  a,  which  is  -r^~7/  »  we 

Ay 

find 

a  =  rh  -T-, 

A 

which  is  the  value  of  a,  corresponding  to  the  asymptotes. 

246.  The  equation  of  the  hyperbola  takes  a  remarkable 
form  when  wre  refer  it  to  the  asymptotes  as  axes.  The 
general  formulas  for  transformation  are 

x  =  x  cos  a  +  y'  cos  a,     y  =  x'  sin  a  +  y'  sin  a'. 

But,  as  the  asymptotes  make  with  the  axis  of  x  angles 

T> 

whose  tangents  are  ±  -r-  >  we  have 

B  B 

tang  a  —  --  -r-  >      tang  a  =  +  -=-  - 

A  A. 

Substituting  the  values  of  x  and  y  in  the  equation  of  the 
hyperbola, 

Ay  —  BV  =  —  A2B2, 
it  becomes 


(A2sinV  —  B2cosV)  ?/'2  +  (A2sin2a—  B2cos2«)  ~)  _ 
x'~  +  2  (A2  sin  a  sin  a'  —  B2  cos  a  cos  a')  x'  y'      5 


CHAP.  IV.] 


ANALYTICAL  GEOMETRY. 


167 


The  co-efficients  of  x'2,  y'2  disappear  in  virtue  of  the  pre 
ceding  values  of  tang  «,  tang  a',  and  that  of  x'  \j  reduces  to 

4A2B2 
—  ~p^  +  ff  »   and  the  equation  of  the  curve  becomes 

A2  +  B2 
xy  =    -  -—  , 


which  is  the  equation  of  the  hyperbola  referred  to  its  asymp 
totes. 

If  we  deduce  the  value  of  y',  we  have 


as  x'  increases  y'  diminishes,  and  when  x  =  x,  y'  =  o,  which 
proves  the  same  property  of  the  asymptotes  continually 
approaching  the  curve,  which  has  been  just  stated. 

247.  If  we  take  the  line  BB'  for  the  transverse  axis  of  the 
hyperbola,  and  AX',  AY',  for  the  asymptotes,  BE  parallel  to 
AX',  will  be  equal  to  VA2  +  B2.  But  BK  drawn  perpen 
dicular  to  BB'  at  B  is  equal  to  AE.  Hence,  AK  =  BE,  and 
AD  =  BD.  As  the  same  thing  may  be  shown  with  respect 
to  the  other  asymptote,  ADBD'  will  be  a  rhombus,  whose 

/A2  -f  B2 
side  AD  =  i  AK=  \J  -          -  .     Let  /3  represent  the  angle 

X'AY'  which  the  asymptotes  make  with  each  other,  the  pre- 


163  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

ceding  equation  of  the  hyperbola  multiplied  by  sin  ft  gives 

,.",-.«       A2  +  B2 

x  y  sm  B  =  -. —  sin  8. 

4 

The  first  member  represents  the  area  of  the  parallelogram 
APMQ,  constructed  upon  the  co-ordinates  AP,  PM,  of  any 
point  of  the  hyperbola;  the  second  member  represents  the 
area  of  the  parallelogram  ADBD',  constructed  upon  the  co 
ordinates  AD',  D'B,  of  the  vertex  B  of  the  hyperbola.  Hence 
the  area  APMQ  is  equivalent  to  that  of  the  figure  ADBD'. 
The  rhombus  BEB'E',  which  is  equal  to  four  times  ADBD',  is 
called  the  Power  of  the  Hyperbola. 

248.  When    the   hyperbola    is    equilateral    A  =  B,  angle 
B  —  90°,  sin/3  — 1,  and  the   rhombus   ADBD'    becomes   a 
square  which  is  equivalent  to  the  rectangle  of  the  co-ordi- 

A2  4-  B2 
nates.    For  more  simplicity,  put ^ —  =  M2,  and  suppress 

the  accents  of  x ,  y' ',  we  shall  have 

xy  —  M2, 
for  the  equations  of  the  hyperbola  referred  to  its  asymptotes. 

249.  Let  it  be  required  to  find  the  equation  of  a  tangent 
line  to  the  hyperbola  referred  to  its  asymptotes. 

Let  x",  y" ,  be  the  co-ordinates  of  the  point  of  tangency. 
They  must  satisfy  the  equation  of  the  hyperbola,  and  hence 
we  have 

x"  y"  =  M2.  (2.) 

The  general  equation  of  the  tangent  line  is 

y-y"  =«<*-*"). 

it  is  required  to  determine  a. 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  189 

Regarding  the  tangent  line  as  a  secant  whose  points  of 
intersection  coincide,  we  have  by  subtracting  equation  (2) 
from 

xy  =  M3, 
the  equation 

xy  —  x"  y"  =  of 
which  may  be  put  under  the  form 

x  (y  —  y")  +  y"  (x  —  x")  =  o. 
Putting  for  y  —  y",  its  value,  we  have 

(x  —  x")  (ax  +  y")  =  o. 
This  equation  is  satisfied  when 
x  —  x"  =  o, 

which  gives  x  =  x"  and  y  =  y",  and  these  values  determine 
the  co-ordinates  of  the  first  point  of  intersection.  Placing 
the  other  factor  equal  to  zero,  we  have 

ax  +  y"  =  o, 
when  the  secant  becomes  a  tangent, 

x  =  x",  and   y  =  y", 
which  gives 

ax"  +  y"  =  o,  or  a  =  —  —  . 

Substituting  this  value  of  a  in  the  equation  of  the  tangent, 
it  becomes 


Making  y  —  o  gives  the  point  in  which  it  cuts  the  axis  of 
x,  and  x  =  x"  will  be  the  subtangent,  which  we  find  to  be 

x  —  x"  =  x", 

that  is,  the  subtangent  is  equal  to  the  abscissa  of  the  points 
15  w 


ANALYTICAL  GEOMETRY. 


[CHAP.  IV 


of  tangency.  To  draw  the  tangent,  take  on  the  asymptote 
a  length  PT  =  AP  =  x",  MT  will  be  the  tangent  required. 
We  see  by  this  construction,  that  if  wre  produce  the  line  MT 
until  it  meets  the  other  asymptote  at  t,  we  shall  have  M£  = 
MT.  The  portion  of  the  tangent  which  is  comprehended 
between  the  asymptotes  is  therefore  bisected  at  the  point  of 
tangency. 

250.  The  equation  of  a  line  passing  through  any  point  M", 
whose  co-ordinates  are  x",  y",  is 


X 


y  —  y"  =  a  (x  —  x"). 

The  other  point  M'"  in  which  this  line 
meets  the  curve,  is  determined  from  the 
equation  (Art.  249), 


ax  +  if  =  o, 


which  gives 


x  =  —  — 


This  is  the  value  of  the  abscissa  AP".     But  if  we  make 
y  =  o  in  the  equation  of  the  straight  line,  it  gives  also 


in  which  x  represents  the  abscissa  AQ"  of  the  point  in  which 
this  line  meets  the  axis  AX,  and  x  —  x"  is  the  value  of  P"Q". 
Hence  P"Q"  =  AP".  Consequently  if  we  draw  M'"Q'  pa 
rallel  to  AX,  the  triangles  P"M"Q",  Q'M'"Q'"  will  be  equal, 
and  the  lines  M"Q",  M'"Q'",  will  be  also  equal  ;  that  is,  if 
through  any  point  of  the  hyperbola,  a  straight  line  be  drawn 
terminated  in  the  asymptotes,  the  portions  of  this  line  compre 
hended  between  the  asymptotes  and  the  curve  will  be  equal 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  171 

251.  This  furnishes  us  with  a  very  simple  method  of 
describing  the  hyperbola  by  points,  when  \ve  know  one 
point  M"  and  the  position  of  its  asymptotes,  for  drawing 
through  this  point  any  line  Q''M"Q,'"  terminated  by  the 
asymptotes,  and  laying  off  from  Q'"  to  M"'  the  distance  Q"M'" 
M"  will  be  a  point  of  the  curve.  Drawing  any  other  line 
through  either  of  these  points,  we  may  in  the  same  way  find 
other  points  of  the  curve.  This  construction  may  also  be 
used  when  we  know  the  centre  and  axes  of  the  hyperbola. 
For  with  these  given,  we  may  easily  construct  the  asymptotes. 


Of  the  Polar  Equation  of  the  Hyperbola,  and  of  the 
Measure  of  its  Surface. 

252.  Resuming  the  equation  of  the  hyperbola  referred  to 
its  centre  and  axes, 

Ay  _  BV  =  —  A2B2, 

we  derive  its  polar  equation,  by  substituting  for  x  and  y 
their  values  dra\vn  from  the  formulas 

x  =  a  +  r  cos  vf 
y  =  b  +  r  sin  v. 

The  substitution  gives 

A2  sin2*;       7  r2  +  2A26  sin  v  ) 

™       2    C      oi>2  tr  +  A*b  —  BV-f  A2B2  =  o, 

—  B2  cos  2v  3  —  2B2a  cos  v    3 

for  the  general  polar  equation  of  the  hyperbola. 

253.  When  the  pole  is  at  one  of  the  foci,  we  have  a  =  d= 
v'A2  -f  B2.     b  =•  o ;  taking  the  positive  value  of  a,  corres- 


172 


ANALYTICAL  GEOMETRY. 


[CttA*.  IV. 


ponding  to  the  point  F,  the  substitution  gives  for  the  two 
values  of  r, 

B  =           B2 

A  —  a  cos  v  A  —  a  cos  v 

If  \ve  make  v  =  o,  the 
radius  vector  takes  the 
position  FX.  Then  cos  v 
=  1,  the  denominator  of/ 
becomes  A  —  a  =  A  — 


N/  A2  +  B2,  a  quantity 
which  is  essentially  nega* 
tive.  Hence  the  curve  has  no  real  points  in  this  direction, 
and  this  will  be  the  case  until  cos  v  is  so  small,  that  the  pro 
duct  a  cos  v  shall  be  less  than  A.  The  condition  will  be 
fulfilled  when  A  +  a  cos  v  —  o,  which  gives 

A  A 

COS  V  —  —  ~    —==-•==:  . 


This  value  of  the  angle  v  is  the  same  wrhich  the  asymptotes 
make  with  the  axis.  The  radius  vector  then  becomes  real, 
and  is  infinite.  For  every  value  of  v  greater  than  this  limit, 
but  less  than  90°,  a  cos  v  is  positive,  and  less  than  A ;  when 
v  >  90°,  a  cos  v  becomes  negative,  and  —  a  cos  v  positive. 
In  this  case  A  —  a  cos  v  is  positive  as  well  as  r.  The  points 
which  this  value  of  r  gives,  correspond  then  to  the  branch 
of  the  hyperbola  situated  on  the  positive  side  of  the  axis  of  x. 

254.  But  in  discussing  the  second  root,  we  shall  see  that 
it  belongs  to  the  other  branch.  In  fact,  it  gives  imaginary 
values  for  all  values  of  the  cos  v  between  the  limits  cos  v  =  1 


and  cos  v  =  - —  —  •     All  the  other  values  of  v  greater  than 


HAP.  IV.]  ANALYTICAL  GEOMETRY.  173 

that  of  the  second  limit  will  give  positive  values  for  r,  and 
when  v  =  180°,  the  radius  vector  will  determine  the  vertex  B'. 

255.  To   put  the  preceding  expressions   under  the  form 
adopted  in  the  ellipse,  make 


a 
e  =  -r-  >     ore  — 


A  A 

in  which  e  represents  the'  ratio  of  the  eccentricity  to  the 
semi-transverse  axis,  and  the  values  of  r  become 


1  —  e  cos  v  1  +  e  cos  v 

These  two  equations  determine  points  situated  on  the  two 
branches  of  the  hyperbola. 

256.  We  have  seen  that  a  similar  transformation  gives 
two  values  for  the  radius  vector  in  the  ellipse,  but  that  one 
of  these  values  is  constantly  negative  and  consequently 
belongs  to  no  point  of  the  curve,  while  for  the  hyperbola  we 
find  two  separate  and  rational  values  for  r,  corresponding  to 
the  two  branches  of  the  hyperbola.  Let  us  examine  this 
difference.  If  in  the  first  of  the  preceding  equations,  we 
count  the  angle  v  from  the  vertex  of  the  curve,  it  will  be 
necessary  to  change  v  into  180°  —  v,  and  we  have  then 

r=      A  (1-6°) 
1  +  e  cos  v 

This  value  of  r  will  equally  give  every  point  of  the  branch 
to  which  it  belongs  by  attributing  suitable  angles  to  v.  But 
operating  in  the  same  way  in  Art.  194  on  the  ellipse,  that  is, 
counting  the  angle  v  from  the  nearest  vertex,  we  get 

_  A  (1  —  e2) 
1  4-  e  cos  v 
15* 


174  ANALYTICAL  GEOMETRY.  [CHAP.  IV. 

This  equation  is  therefore  absolutely  the  same  for  the  two 
cases,  only  in  the  ellipse  e  is  less  than  unity,  wrhile  it  is 
greater  in  the  hyperbola.  Besides,  the  sign  of  A  is  changed, 
Let  us  now  make  e  =  I  and  A  =  infinity,  we  shall  have, 
making  A  (1  —  e2)  —  p, 

1  +  cos  v 

which  is  the  polar  equation  of  the  parabola.     Hence  we  see 
that  the  equation 

_  A  (1  — •  e2) 
1  +  e  cos  v 

may  in  general  represent  all  the  conic  sections,  by  giving 
suitable  values  to  A  and  e. 

257.  We  may  deduce  the  equation  of  the  hyperbola  in  the 
same  manner  as  we  have  that  of  the  ellipse  in  Art.  196,  by 
introducing  one  of  its  properties  which  characterize  it.     The 
method  being  similar  to  that  of  the  ellipse,  it  will  be  unne 
cessary  to  repeat  it  here. 

258.  We  have  seen  that  the  equilateral  hyperbola  bears 
the  same  relation  to  other  hyperbolas  that  the  circle  does  to 
the  ellipse.     In  applying  here  what  has  been  said  (Art.  215), 
we  may  compare  a  portion  of  any  hyperbola,  to  the  corres 
ponding  area  of  an  equilateral  hyperbola  having  the  same 
transverse  axis,  and  there   results    that  the-se   are   to  each 
other  in  the  ratio  of  the  conjugate  axes.     The  absolute  areas 
however  can  only  be  obtained  by  means  of  logarithms. 

259.  We  have  found  (Art.  156)  for  the   equation  of  the 
Ellipse  referred  to  its  vertex, 

B2 

y  -  -  A* 


CHAP.  IV.]  ANALYTICAL  GEOMETRY.  175 

for  the  equation  of  the  parabola,  we  have 

if  =  2px, 
and  for  the  hyperbola 


These  equations  may  all  be  put  under  the  form 
if  =  mx  +  ntf, 

in  which  m  is  the  parameter  of  the  curve,  and  n  the  square 
of  the  ratio  of  the  semi-axes. 

In  the  ellipse  n  is  negative,  in  the  hyperbola  it  is  positive, 
and  in  the  parabola  it  is  zero. 


176  ANALYTICAL  GEOMETRY.  [CHAP.  V. 


CHAPTER    Y. 

DISCUSSION  OF  EQUATIONS. 

260.  HAVING  discussed  in  detail  the  particular  equations  of 
the  Circle,  Ellipse,  Parabola,  and  Hyperbola,  we  will  apply 
the  principles  which  have  been  established  to  the  discussion 
of  the  general  equation  of  the  second  degree  between  two 
indeterminates. 

Let  us  take  the  general  equation 

A?/2  +  Bxy  +  Car2  +  Dy  +  Ex  +  F  =  o, 

in  which  x  and  y  represent  rectangular  co-ordinates.  Let 
us  seek  the  form  and  position  of  the  curves  which  it  repre 
sents,  according  to  the  different  values  of  the  independent 
coefficients  A,  B,  C,  D,  E,  F.  Resolving  this  equation  with 
respect  to  y,  we  have 

B*  +  D  1  /  (B2— 4  AC)  x2 + 2(BD— 2  AE)* + D2— 4  AF 
~2A~±  2A  V 

In  consequence  of  the  double  sign  of  the  radical,  there 
will,  in  general,  be  two  ordinates  corresponding  to  the  same 
abscissa,  which  we  may  determine  and  construct,  if  the  values 
given  to  x  cause  the  radical  to  be  real.  If  they  reduce  it  to 
zero,  there  will  be  but  one  value  of  y,  and  if  they  render  it 
imaginary,  there  will  be  no  point  of  the  curve  corresponding 
to  these  abscissas. 

Hence,  to  determine  the  extent  of  the  curve  in  the  direc- 


CHAP.  V.]  ANALYTICAL  GEOMETRY.  177 

tion  of  the  axis  of  x,  we  must  seek  whether  the  values  given 
to  x  render  the  radical,  real,  zero,  or  imaginary. 

261.  In  this  discussion  we  will  suppose  that  the  general 
equation  contains  the  second  power  of  at  least  one  of  the 
variables  x  or  y.  For,  if  the  equation  were  independent  of 
these  terms,  its  discussion  would  be  rendered  very  simple, 
and  the  curve  which  it  represents  immediately  determined. 
The  general  equation  under  this  supposition  would  reduce  to 

Exy  +  Dy  +  Ex  +  F  =  o, 
which  may  be  put  under  the  form 


and  making 

D  E 

it  becomes 

DE  — BF 
B2 


which  is  the  equation  of  an  hyperbola  referred  to  its  asymp 
totes  (Art.  246). 

262.  The  result  would  be  still  more  simple  if  the  coeffi 
cients  A,  B,  C,  reduced  the  three  terms  in  x*,  ]f,  and  xy,  to 
zero.  In  this  case  the  general  equation  would  become  of  the 
first  degree,  and  would  evidently  represent  a  straight  line, 
which  could  be  readily  constructed.  These  particular  cases 
presenting  no  difficulty,  we  will  suppose  in  this  discussion 
that  the  square  of  the  variable  y  enters  into  the  general 
equation. 

X 


178  ANALYTICAL  GEOMETRY.  [CHAP.  V. 

263.  Resuming  the  value  of  y  deduced  from  the  general 
equation, 


Ex  +  D      J_      /(B2—  4AC)*2+2(BD—  2AE)*+D2—  4AF 

2A~    ~  2A  V 

we  see  that  the  circumstances  which  determine  the  reality 
of  y  depend  upon  the  sign  of  the  quantity  under  the  radical. 
But  we  know  from  Algebra,  that  in  an  expression  of  this 
kind,  we  can  always  give  such  a  value  to  x,  as  to  make  the 
sign  of  this  polynomial  depend  upon  that  of  the  first  term: 
and  since  x*  is  positive  for  all  real  values  of  x,  the  'sign  will 
depend  upon  that  of  the  quantity  (B2  —  4AC).  We  may 
therefore  conclude, 

1st.  When  B2  —  4AC  is  negative)  there  will  be  values  of  x 
both  positive  and  negative,  for  which  the  values  of  y  will  be 
imaginary.  The  curve  is  therefore  limited  on  both  sides  of 
the  axis  of  y. 

2dly.  When  (B2  —  4AC)  =  o,  the  first  term  of  the  poly 
nomial  disappears,  and  the  sign  of  the  polynomial  will  then 
depend  upon  that  of  the  second  term  (BD  —  2AE)  x.  If 
(BD  —  2AE)  be  positive,  the  curve  will  extend  indefinitely 
for  all  values  of  x  that  are  positive.  But  if  x  be  negative,  y 
becomes  imaginary.  The  curve  is  therefore  limited  on  the 
side  of  the  negative  abscissas.  The  reverse  will  be  the  case 
if  (BD  —  2AE)  is  negative.  The  curve  will  in  this  case 
extend  indefinitely  when  x  is  negative,  and  be  limited  for 
positive  values  of  x. 

3dly.  When  (B2  —  4AC)  is  positive,  there  will  be  positive 
and  negative  values  for  a?,  beyond  which  those  of  y  will  be 
always  real.  The  curve  will  therefore  extend  indefinitely  in 
both  directions. 


CHAP.  V.]  ANALYTICAL  GEOMETRY.  179 

264.  \Ye  are  therefore  led  to  divide  curves  of  the  second 
order  into  three  classes,  to  wit, 

1.  Curves  limited  in  every  direction; 

Character,  .  .  .  B2  —  4AC  <  o. 

2.  Curves  limited  in  one  direction,  and  indefinite  in  the 
opposite ; 

Character,  .  .  .  B2  —  4AC  =  o. 

3.  Curves  indefinite  in  all  directions ; 

Character,  .  .  .  B2  —  4AC  >  o. 

The  ellipse  is  comprehended  in  the  first  class,  the  parabola 
in  the  second,  and  the  hyperbola  in  the  third.  We  \vill  dis 
cuss  each  of  these  classes. 

FIRST  CLASS. — Curves  limited  in  every  direction. 

Analytical  Character,  B2  —  4AC  <^o. 

265.  Let  us  resume  the  general  value  of  y, 

y  =  — 

Bx  +  D      _1_      /(B2—  4ACX+2(BD— 2AE)*+D2—  4AF. 
2A          2A  V 

This  expression  shows,  that,  to  find  points  in  the  curve 
we  must  construct  for  every  abscissa  AP  an  ordinate  equal 

to  —   <  — g  .  ,     >  which  will  determine  a  point  N,  above 

and  below  which  we  must  lay  off  the 
distance  represented  by  the  radical. 
From  which  it  follows  that  each  of 
the  points  N  bisects  the  corresponding 
line  MM',  which  is  limited  by  the 

C  Ex  +  D  7 

curve.  This  quantity  —  •<  — ^-r —  S- 

/       &A.        \ 


180  ANALYTICAL  GEOMETRY.  [CHAP.  V, 

which  varies  with  x,  is  the  ordinate  of  a  straight  line  whose 
equation  is 

v  =  —  <     X     —  >  • 
y  ^      QA      \ 

This  line  is,  therefore,  the  locus  of  the  points  N,  which  we 
have  just  considered.  Hence,  it  bisects  all  the  lines  drawn 
parallel  to  the  axis  of  y  and  limited  by  the  curve.  This  line 
is  called  the  diameter  of  the  curve. 

266.  Let  us  now  determine  the  limit  of  the  curve  in  the 
direction  of  the  axis  of  x.  For  this  purpose  we  may  put  the 
polynomial  under  the  radical  under  another  form, 


and   if  we   represent   by  x'  and   x"  the   two  roots   of  the 
equation 

2        BD  —  2AE        D2  —  4AF  _ 

L2  B2  —  4AC  x  F  B^  = 

the  value  of  y  will  take  the  form 


Ba?  +  D         1        / 
~          ~~A~~       2A  V 


(B2  —  4AC)  (x  —  x')  (x  —  x"). 


Hence  we  see,  the  values  of  y  will  be  real  or  imaginary 
according  to  the  signs  of  the  factors  (x  —  x')  and  (x  —  x"), 
and  consequently,  the  limits  of  the  curve  will  depend  upon 
the  values  of  x'  and  x".  These  values  may  be  real  and  un 
equal,  real  and  equal,  or  imaginary.  We  will  examine  these 
three  cases. 

267.  1st.  If  the  roots  are  real  and  unequal,  all  the  value 


CHAP.  Y.J  ANALYTICAL  GEOMETRY.  181 

of  x  greater  than  x'  and  less  than  x" ',  will  give  contrary 
signs  to  the  factors  x  —  x',  x  —  x",  and  this  product  will  be 
negative,  but  as  B2  —  4AC  is  also  negative,  the  quantity 
(B2  —  4AC)  (x  —  x')  (x  —  x")  will  be  positive,  and  the  ordi- 
nate  y  will  have  two  real  values.  If  we  make  x  =  x'  or 
x  =  x",  the  radical  will  disappear,  the  two  values  of  y  will 

be  real  and  equal  to ^r- —  •     In  this  case  the  abscissas 

x'  and  x"  belong  to  the  points  in  which  the  curve  meets  its 
diameter,  that  is,  to  the  vertices  of  the  curve.  Finally,  for 
x  positive  or  negative,  but  greater  than  x'  and  x",  the  two 
"actors  (x  —  x'),  (x  —  x"),  will  have  like  signs,  and  their 
product  (x  —  a:')  (x  —  x")  will  be  positive;  and  since  B2  — 
4AC  is  negative,  the  quantity  (B2  —  4AC)  (x  —  x')  (x  —  x"). 
will  be  negative  also,  and  both  values  of  y  will  be  imaginary. 

268.  We  see  from  this  discussion  that  the  curve  is  con 
tinuous  between  the  abscissas  x',  x",  but  does  not  extend 
beyond  them ;  and  if  at  their  extremities  we  draw  two  per 
pendiculars  to  the  axis  of  x,  these  lines  will  limit  the  curve, 
and  be  tangent  to  it,  since  we  may  regard  them  as  secants 
whose  points  of  intersection  have  united. 

269.  By  resolving   the   equation  with    respect   to   x,  we 
would    arrive  at  similar  conclusions,  and  the  limits  of  the 
curve  in  the  direction  of  the  axis  of  y,  would  be  the  tangents 
to  the  curve  drawn  parallel  to  the  axis  of  x. 

270.  Having  thus  found  four  points  of  the  curve,  we  could 
ascertain  the  points  in  which  the  curve  cuts  the  co-ordinate 
axes.     By  making  x  =  o,  we  have 

A?/2  +  Dy  +  F  =  o, 

and  the  roots  of  this  equation  will  give  the  points  in  which 
16 


182  ANALYTICAL  GEOMETRY.  [CHAP.  T 

the  curve  cuts  the  axis  of  y.  According  as  the  values  of  y 
are  real  and  unequal,  real  and  equal,  or  imaginary,  the  curve 
will  have  two  points  of  intersection  with  the  axis  of  y,  be 
tangent  to  it,  or  not  meet  it  at  all. 

271.  By  making  y  —  o,  we  have 

Or2  +  Ex  +  F  =  o, 

and  the  roots  of  this  equation  will  in  the  same  manner  deter 
mine  the  points  in  which  the  curve  cuts  the  axis  of  x. 

272.  In  comparing  this  curve  with  those  of  the  Conic 
Sections,  we   see  at  once  its  similarity  to  the  Ellipse.     Its 
position  will  depend  upon  the  particular  values  of  the  co 
efficients  A,  B,  C,  &€. 

273.  Let  us  apply  these  principles  to  a  numerical  example, 
and  construct  the  curve  represented  by  the  equation 


yz 
In  this  example  we  have 

A=  1,  B  =  —  2, 

hence 

B2  —  4AC  =  4  —  8  <  o. 

The  curve  which  this  equation  represents  belongs  to  the 
first  class  of  curves,  which  corresponds,  as  we  shall  presently 
gee,  to  the  Ellipse. 

Resolving  this  equation  with  respect  to  y,  we  have 


y  =  (x  +  1)  =fc  V  (x  +  I)2  —  2*  (x  +  1) 
The  equation 

y  =  (x  +  l), 


CHAP.  V.]  ANALYTICAL  GEOMETRY.  183 

is  that  of  the  diameter  of  the  curve,  and  laying  off  on  the 

axis  of  y  a  distance  AB  equal    to 

1,  and  drawing  BC  making  an  angle 

of  45°  with  the  axis  of  x,  BC  will 

be  this  diameter.     The  roots  of  the 

equation 


are 


ar= 


a?  =  —  I. 


Laying  off  on  both  sides  of  the  axis  of  y  distance  AC  and 
AD  equal  to  1,  the  perpendiculars  CP,  DP',  will  limit  the 
carve  in  this  direction.  Substituting  the  values  of  x  in  the 
original  equation,  we  have  the  corresponding  values  of  y, 

y  =  +  2,    y  =  o. 

The  first  gives  the  point  P',  the  second  the  point  C. 
Making  x  =  o,  the  equation  becomes 


which  gives 

y  =  o,    y=  +2, 

for  the  points  A  and  H,  in  which  the  curve  cuts  the  axis  of  y. 
For  y  —  o,  we  have 

z*  +  x  =  o, 
and 

x  =  o,    x  =  —  1, 

corresponding  to  the  points  A  and  C  on  the  axis  of  x. 

274.  The  following  examples  may  be  discussed  in  the  same 
manner  : 


184 


ANALYTICAL  GEOMETRY. 


[CHAP.  V. 


2.    2  — 


—   x  =  o. 


3.  t2  —  %x    -t-  2^  +  2y  +  a;  +  3  =  o. 


275.  There  is  a  particular  case  comprehended  under  this 
class,  which  it  would  be  well  to  examine.  It  is  that  in  whicn 
A  =  C  and  B  —  o  in  the  general  equation.  This  supposition 
gives  , 

Ay2  +  Ax2  +  Dy  +  Ex  +  F  =  o  ; 


or  dividing  by  A, 


D         E 


D2  +  E2 

If  we  add  —  TT2~   to  k°tn  sides  of  this  equation,  it  may 


be  put  under  the  form 


E 


—  4AF 


4Aa 


CHAP.  Y.I  ANALYTICAL  GEOMETRY.  185 

If  the  co-ordinates  x,  y,  are  rectangular,  this  equation  is  of 
the  same  form  as  that  in  Art.  139,  and  therefore  represents  a 

D  E 

circle,  the  co-ordinates  of  whose  centre  are  —  TTT->   —  ^-r  , 

2A          2A 


r      .    VD2  +  E2  — 4AF      . 

and  whose  radius  is    -rr •     In  order  that  this 

ZA. 

circle  be  real,  it  is  necessary  that  the  quantity  (D2  +  E2  — 
4AF)  be  positive.  If  D2  +  E2  —  4AF  =  o,  the  circle  reduces 
to  a  point.  If  the  system  of  co-ordinates  be  oblique,  this 
equation  will  be  that  of  an  ellipse. 

276.  We  now  come  to  the  second  supposition,  in  which 
the  roots  x',x"9  are  equal.  The  product  (x  —  x')  (x — x") 
becomes  (x  —  x')2,  and  the  general  value  of  y  is 


Ex  +  D      x  —  x 


Whatever  value  we  give  to  x  which  does  not  reduce  x  —  xf 
to  zero,  will  give  imaginary  values  for  y,  since  B2  —  4AC  is 
negative.  But  if  x  =  x',  there  will  be  but  one  value  for  y, 

which  will  be  real  and  equal  to  —    <  — -—    >  •     In  this 

case  the  curve  reduces  to  a  single  point,  situated  on  the 
diameter,  the  co-ordinates  of  which  are 

EXAMPLES. 

x*  +  y2  =  °>   y*  +  ff2  —  2r  +  i  =  o. 

277.  Finally,  when  the  roots  are  imaginary.     In  this  case 
the  product  (x  —  x')  (x  —  x")  will  always  be  positive,  what- 
16* 


186  ANALYTICAL  GEOMETRY.  [CHAP.  V. 

ever  value  be  given  to  x.     For  the  roots  x  ',  and  x'\  are  of  the 
form 

x'  =  rbp  +  q  V—  1, 


hence, 

(a  —  a?1)  (a;  —  a:")  =  x2  ±  %w?  +  p'  +  g"  =  (a? 

which  last  expression  is  always  positive  for  any  real  value  of 
x.  The  product  (a?  —  x1)  (x  —  a?")  being  positive,  and  (B2  — 
4AC)  negative,  the  quantity  under  the  radical  is  negative, 
and  the  values  of  y  become  imaginary.  There  is  therefore 
no  curve. 

EXAMPLES. 

y2  +  xy  +  x2  +  1*  +  y  +  1  =  o,     y2  +  x*  +  2a?  +  2  =  o, 
which  may  be  put  under  the  forms  respectively 

(2y  +  x  +  I)2  +  3^2  +  3  =  o,     y2  +  (x  +  I)2  +  1  =  o. 

278.  There  results  from  the  preceding  discussion,  that  the 
curves  of  the  second  order,  comprehended  in  the  first  class, 
for  which  B2  —  4AC   is  negative,  are  in  general  re-entrant 
curves  as  the  ellipse,  but  the  secondary  conditions  give  rise 
to  three  varieties,  which  are  the  Point,  the  Imaginary  Curve, 
and  the  Circle. 

SECOND  CLASS.  —  Curves  limited  in  one  direction  and  indefinite 
in  the  opposite. 

Analytical  Character,  B2  —  4  AC  =  o. 

279.  In  this  case  the  general  value  of  y  becomes 


2(BD  —  2AE)a?  +  D2  —  4AF. 


CHAP.  V.]  ANALYTICAL  GEOMETRY.  187 

Making,  for  more  simplicity, 

D2  — 4AF 

—  — —  x  f 


2(BD  —  2AE) 
it  may  be  put  under  the  form 


Bx  +  D      J_      /  2  (BD  —  2AE)  (x  —  x1). 
y~         ~2A~      2A  V 

If  BD  —  2AE  is  positive,  so  long  as  x  is  greater  than  x', 
the  factor  x  —  x'  will  be  positive,  and  the  radical  will  be 
real.  If  x  =  x',  the  radical  will  disappear,  and  if  #  be  Jess 
than  x,  the  factor  x  —  x'  will  be  negative,  and  the  radical 
will  be  imaginary.  The  curve  therefore  extends  indefinitely 
from  x  =  x'  to  x  =  +  infinity.  The  ordinate  corresponding 
to  x  =  x',  will  be  tangent  to  the  curve  at  this  point. 

280.  The  contrary  will  be  the  case  if  BD  —  2AE  is  nega 
tive.  The  curve  will  extend  indefinitely  on  the  side  of  the 
negative  abscissas,  and  will  be  limited  in  the  opposite 
direction. 

In  both  cases  the  straight  line  whose  equation  is 


will  be  the  diameter  of  the  curve. 


EXAMPLES. 


=  0. 


188 


ANALYTICAL  GEOMETRY.  [CIIAP.  V. 


A 


2.   /*  — 


3.  tf  —  Zxy  +  3*  +  2#  +  I  s  9. 


4.  jf  — 

—  1=0. 


CHAP.  V.I 


ANALYTICAL  GEOMETRY. 


189 


5.  if—Zxy  +  x2  — 
—  2r  =  o. 


281.  If  BD  —  2AE  =  o,  the  value  of  y  becomes 
Bx  +  D  )       _1_      7D2  —  4AF. 

The  curve  becomes  two  parallel  straight  lines,  which  will 
be  real,  one  straight  line,  or  two  imaginary  lines,  according 
as  D2  —  4AF  is  positive,  nothing,  or  negative. 

EXAMPLES. 


if  — 


—     =o. 


\ 


190 


ANALYTICAL  GEOMETRY. 


[CHAP.  T 


3.  if—  Zxy  +  of  +  2y  —  2*  f  I  =  o 


4.  y2  —  4xy  -f  4ff*  =  o, 


5.  if  +  2xy  +  s?  —  1  =  o 

6.  y*  +  y  +  1  =  o. 


282.  There  results  from  this  discussion,  that  the  curves  of 
the   second  order,  comprehended   in    the   second   class,  foi 
which  B2  —  4AC  =  o,  are  in  general  indefinite  in  one  direc 
tion,  as  the  parabola,  but  include  as  varieties  two  parallel 
straight  lines,  one  straight  line,  and  two  imaginary  straight 
lines. 

THIRD  CLASS. — Curves  indefinite  in  every  direction. 
Analytical  Character,  B2  —  4AC  >  o. 

283.  The  discussion  of  this  class  of  curves  presents  no 
difficulty,  as  the  method  is  precisely  similar  to  that  of  the 
first. class.     Resuming  the  general  value  of  y, 


CHAP.  V.]  ANALYTICAL  GEOMETRY. 

B.r-f  D        1 


101 


BD  —  2AE         D2  —  4AF 

a    ~ 


and  representing  by  x  ',  x",  the  roots  of  the  equation 
BD  —  2AE      D2—  4AF 


B2—  4AC 


the  value  of  y  becomes 


2A 


'.2A 


So  long  as  x'  and  x"  are  real,  the  curve  will  be  imaginary 
between  the  limits  a?',  x",  since  (B2  —  4AC)  is  positive,  but 
for  all  values  of  #,  positive  as  well  as  negative,  beyond  this 
limit,  the  values  of  y  will  be  real.  The  abscissas  x',  x",  cor 
respond  to  the  points  in  which  the  curve  intersects  its  dia 
meter:  and  the  equation  of  this  diameter  is, 

Bx  +  D 


EXAMPLES. 


I.    2  —  2r   —  x8  4-2  =  0. 


\ 
/ 
I 


192 


ANALYTICAL  GEOMETRY.  [CHAP.  V 


A 


2.  t2—- 


=  o. 


\ 


3  =  o. 


4.  if  —  2^— %  +  60:  —  3  = 


ANALYTICAL  GEOMETRY. 


193 


CHAP.  V.] 

284.  We  may  find  the  points  in  which  the  curve  cuts  the 
axes  by  the  methods  pursued  in  Arts.  287  and  288. 

285.  When  the  roots  x'y  x",  are  equal,  the  product  (#  —  a:') 
(x  —  x")  would  reduce  to  (a?  — a/)2,  and  we  would  have 


2A 


x  —  x' 
:~~2A~ 


2  — 4AC. 


This   equation   represents  two  straight   lines,  which   are 
always  real,  since  B2 — 4AC  is  positive. 


EXAMPLES. 


i,  y — a*  +  %  + 1 


L 


\l 


3.  y2  +  xy  —  2r*  +  3*  —  1  =  o. 


17 


194  ANALYTICAL  GEOMETRY.  [CHAP.  V, 

286,  When  x'  and  x"  are  imaginary,  the  quantity  under 
the  radical  will  be  always  positive,  since  (x  —  x')  (x  —  x")  is 
positive,  whatever  value  be  given  to  x  (Art.  293),  and  B2  — 
44-C  is  positive  for  this  class  of  curves.  Hence,  whatever 
value  we  give  to  x,  that  of?/  will  be  real,  and  will  give  points 
of  the  curve.  This  curve  will  be  composed  of  two  separate 
branches,  and  the  line  represented  by  the  equation 

Ex  —  D 


y  =  — 


2A 


will  be  its  diameter. 


As  the  radical  V(B2—  4AC)  (x  —  xr)  (x  —  x"}  can  never 
reduce  to  zero,  this  diameter  does  not  cut  the  curve. 


EXAMPLES. 


1.       —  2x    —  x*  —  2  =  o. 


—x*  -f 


CHAP.  V.I 


ANALYTICAL  GEOMETRY. 


195 


3.    »--2x— **— 2ar— 2  = 


287.   If  A 
becomes 

or, 


—  C,   and   B  =  o,   the   general   equation 

-—  Ax2  +  Dy  +  Ex  +  F  =  o, 
D        E        F 


which  may  be  put  under  the  form 

D,2      /         Ev2     D2—  E2  —  4AF 


Hence  we  see,  that  if  the  co-ordinates  x  and  y  are  rectan 
gular,  this  equation  represents  an  equilateral  hyperbola,  the 

D         E 

co-ordinates  of  whose  centre  are  —  rrr  '  +  s~r  »  and  whose 

2A        2A 

D2  —  E2  —  4AF 

power  is  -    --  jT2  -  •     This  case  is  analogous  to  that 

of  the  circle  (Art.  291). 

288.  We  conclude  from  this  discussion  that  the  curves  of 
the  second  order,  comprehended  in  the  third  class,  for  which 
B2  —  4AC  is  positive,  are  always  curves  composed  of  two 
separate  and  infinite  branches,  as  the  hyperbola,  and  that 
they  include,  as  varieties,  two  straight  lines  and  the  equilateral 
hyperbola. 


196  ANALYTICAL  GEOMETRY.  [CHAP.  V. 


GENERAL   EXAMPLES. 


1.  Construct  the  equation 

So;2  -f       —  4x  —  3  =  o. 


2.  Construct  the  equation 

__2__2a;2  —  4   —  x+  10  =  o. 


3.  Construct  the  equation 

2*2  —  2*  +  4  =  o. 


4.  Construct  the  equation 

^  __  fay    +     5^    +    2tf    +     1     -    0. 

5.  Construct  the  equation 

%y*  —  %xy  —  x*  +  y  +  4#  —  10  =  o. 

6.  Construct  the  equation 


x  +    y  —    a;  —     =  o. 

7.  Construct  the  equation 

?/2  +  Zxy  +  x9  —  6y  +  9  =  o. 

8.  Construct  the  equation 

a:2  —  2   —  4a?  +  10  =  a 


9.  Construct  the  equation 

^  —  2^  + 

10.  Construct  the  equation 


4-  1  =  o. 


CHAP.  V.]  ANALYTICAL  GEOMETRY.  197 


Of  the  Centres  and  Diameters  of  Plane  Curves. 

289.  The  centre  of  a  curve  is  that  point  through  which,  if 
any  line  be  drawn   terminated  in  the  curve,  the  points  of 
.ntersection  will  be  equal  in  number,  and  the  line  will  be 
bisected  at  the  centre. 

290.  If  we  suppose  this  condition  satisfied,  and  that  the 
origin  of  co-ordinates  is  transferred  to  this  point,  then  it  fol 
lows,  that  if  -f  x',  +  y',  represent  the  co-ordinates  of  one  of 
the  points  in  which  the  line  drawn  through  the  centre  inter 
sects  the  curve,  the  curve  will  have  another  point,  of  which 
the  co-ordinates  will  be  — x',  — y',  that  is,  its  equation  will 
be  satisfied  when  — x',  — y',  are  substituted  for   -f  x',  +  y' 
This  condition  \vill  evidently  be  fulfilled  if  the  equation  of 
the  curve  contain  only  the  even  powers  of  the  variables  x 
and  y,  for  these  terms  will  undergo  no  change  when  — x'  is 
substituted   for    -f  x',  and   —  y'  for    +  y'.     To   determine, 
therefore,  whether  a  given  curve  has  a  centre,  we  must  ex 
amine  if  it  have  a  point  in  its  plane,  to  which,  if  the  curve 
be    referred  as  the  origin  of  co-ordinates,  the  transformed 
equation  will  contain  variable  terms  of  an  even  dimension 
only. 

291.  For  example,  to  determine  whether  curves   of  the 
second  order  represented  by  the  general  equation 

Ay2  -f  Exy  +  Cx2  +  Vy  +  E.r  +  F  =  o, 

have  centres,  we  must  substitute  for  x  and  y,  expressions  of 
the  form 

x  =  a  +  x',    y  =  b  +  y', 

in  which  a  and  b  are  the  co-ordinates  of  the  new  origin,  and 
IT* 


198  ANALYTICAL  GEOMETRY.  [CHAP.  V. 

see  whether  we  can  dispose  of  these  quantities  in  such  a 
manner  as  to  cause  every  term  of  an  uneven  dimension  to 
disappear  from  the  transformed  equation.  If  this  substitu 
tion  be  made,  the  transformed  equation  will  generally  con 
tain  two  terms  of  an  uneven  dimension,  to  wit,  (2A6  +  Ba 
+  D)  y'  and  (2C«  +  B6  +  E)  x'.  And  in  order  that  these 
terms  disappear,  a  and  /;  must  be  susceptible  of  such  values 
as  to  make 

2A6  +  Ba  +  D  =  o,    2Ca  +  Eb  +  E  =  o, 
and  then  the  equation  referred  to  the  new  origin  becomes 
Ay'2  +  B*y  +  Cx'2  +  A62  +  Eab  +  Co2  +  Db  +  Ea  +  F  =  o; 

and  under  this  form  we  see  that  it  undergoes  no  change 
when  —  x',  —  y',  are  substituted  for  +  x',  +  y'. 

292.  The  relations  which  exist  between  the  co-ordinates 
a  and  b  are  of  the  first  degree,  and  represent  two  straight 
lines.  These  lines  can  only  intersect  in  one  point.  Hence, 
curves  of  the  second  order  have  only  one  centre. 

In  fact   these  equations   give   for  a  and  b,  the  following 

values, 

2AE  —  BD  2CD  —  BE 


_ 

''  ~ 


and  these  values  are  single.  They  become  infinite  when 
32  —  4AC  =  o,  which  shows  that  there  is  no  centre,  or  that 
it  is  at  an  infinite  distance  from  the  origin,  which  is  the  case 
with  curves  of  the  second  class.  Here  the  two  lines  whose 
intersection  determines  the  centre  become  parallel.  If  one 
of  the  numerators  be  zero  at  the  same  time  with  the  denomi 
nator,  the  values  of  a  and  b  become  indeterminate.  This 
arises  from  the  fact,  that  this  supposition  reduces  the  two 
equations  to  a  single  one,  which  is  not  sufficient  to  determine 


CHAP.  T.]  ANALYTICAL  GEOMETRY.  199 

two  unknown  quantities.     For  if  we  suppose 

2AE  —  BD  =  a, 
and  B2  —  4AC  =  o, 

we  have  from  the  first  equation 

2AE 

-D" 

which   value  being   substituted  in  the  second  equation,  it 
becomes 

AE2  — D2C  =  o, 
hence 

AE 
'  DT' 

Substituting  this  value  of  C  in  the  numerator  of  the  value 
of  b,  it  becomes  after  reduction 

2AE  —  BD, 

which  is  the  same  expression  as  the  numerator  of  the  value 
of  a. 

The  two  equations  thus  reducing  to  one,  are  not  sufficient 
to  make  known  the  values  of  a  and  b,  and  are  consequently 
indeterminate.  There  are  therefore  an  infinite  number  of 
centres  situated  on  the  same  straight  line.  But  when  BD  — 
2AE  =  o,  and  B2  —  4AC  =  o,  the  curve  reduces  to  two  par 
allel  straight  lines  (Art.  297),  and  all  the  centres  are  found 
on  a  line  between  the  two. 

293.  The  diameter  of  a  curve  is  any  straight  line  which 
bisects  a  system  of  parallel  chords.  If?  therefore,  we  take  a 
diameter  for  the  axis  of  a?,  and  take  the  axis  of  y  parallel  to 
the  chords  which  are  bisected  by  this  diameter,  the  trans 
formed  equation  will  be  such,  that  if  it  be  satisfied  by  the 


200  ANALYTICAL  GEOMETRY.  [CHAP.  V, 

values  -f  x',  +  y',  it  must  also  be  by  +  x',  —  ?/',  that  is,  by 
the  same  ordinate  taken  in  an  opposite  direction.  Conse 
quently,  to  ascertain  whether  a  curve  has  one  or  more 
diameters,  we  must  change  the  direction  of  the  axes  by 
means  of  the  general  formulas 

x  =  a  +  x'  cos  a  +  y'  cos  a',    y  =  b  +  oc'  sin  a  -f  y'  sin  a', 

and  after  substituting  these  values  we  must  determine  a,  b 
a,  a',  in  such  a  manner,  that  all  the  terms  affected  with  un 
even  powers  of  one  of  the  variables  disappear,  without  the 
variables  themselves  ceasing  to  be  indeterminate.  If  this  be 
possible,  the  direction  of  the  other  variable  will  be  a  diameter 
of  the  curve. 

294.  Let  us  apply  these  principles  to  the  general  equation 

A?/2  +  Exy  +  Co:2  +  D?/  +  Ex  +  F  =  o. 

Making  the  substitutions,  we  shall  find,  that  the  transformed 
equation  will  generally  contain  three  terms,  in  which  one  of 
the  variables  x,  y',  will  be  of  an  uneven  degree,  and  these 
terms  are 

J2A  sin  a  sin  a'  +  B  (sin  a  cos  a'  +  sin  a'  cos  a)  + 

2C  COS  a  COS  a'  |  x'y', 

+  j(2A6  +  Ba  +  D)  sin  a  +  (2C#  +  B6  +  E)  cos  a\x 
+  Ba  +  D)sina'+  (2Ca  +  Eb  +  E)  cos  a'y. 


Now,  if  we  wish  to  render  x  a  diameter,  the  co-efficients 
of  the  terms  in  y'  must  disappear,  which  requires  that  w« 
make 

J2A  sin  a  sin  a'  +  B  (sin  a  cos  a'  -f  sin  a'  cos  a)  +  2C  cos  a 
cos  a'  j  a?y  =  o  ; 

or,  what  is  the  same  thing, 


CHAP.  V.]  ANALYTICAL  GEOMETRY.  201 

2C  +  B  (tang  a!  +  tang  a)  +  2A  tang  a  tang  a'  =  o,     (1) 
and  that  \ve  also  have 
5  (2A6  +  Ba  +  D)  sin  a'  +  (2Ca  +  BZ>  +  E)  cos  a'  j  y'  =  o.  (2) 

If,  on  the  contrary,  we  wished  the  axis  of  y'  to  be  a  diam 
eter,  the  co-efficients  of  the  terms  in  x  must  disappear.  But 
this  supposition  would  also  require  equation  (1)  to  be  satisfied 
and  that,  in  addition  to  this,  we  have 

j(2A6'+  Ba  +  D)  sin  a  +  (2Ca  +  Eb  +  E)  cos  aj  x'  =  o.  (3) 

295.  Let  us  examine  what  these  equations  indicate. 
We  see  in  the  first  place,  that  whichever  axis  we  select  for 

a  diameter,  equation  (1)  must  always  exist,  and  it  is  also 
necessary  to  connect  with  it  one  of  the  equations  (2)  or  (3). 
The  first  equation  determines  the  relation  between  a  and  a', 
and  when  one  of  them  is  given,  it  assigns  a  real  value  to  the 
other.  But  after  this  equation  is  thus  satisfied,  the  second 
equation  (2)  or  (3)  which  is  connected  with  it,  can  only  be 
fulfilled  by  giving  proper  values  to  a  and  b  ;  so  that  while 
equation  (1)  assigns  a  direction  to  the  chords  which  are 
bisected  by  the  diameter,  equation  (2)  or  (3)  between  a  and 
b,  will  be  the  equation  of  this  diameter  relatively  to  the  first 
co-ordinate  axes. 

296.  Equations  (2)  and  (3)  are   evidently  both   satisfied 
when  we  make 

2A6  +  Btf  +  D  =  o,        2C<z  +  Eb  +  E  =  o.      (4) 

Hence  the  values  of  a  and  b  given  by  these  conditions 
belong  to  a  point  which  is  common  to  every  diameter.  But 
these  conditions  are  the  same  as  those  which  determine  the 
centre  (Art.  307). 

2A 


202  ANALYTICAL  GEOMETRY.  [CHAP.  V. 

Hence  every  diameter  of  curves  of  the  second  order  passes 
through  the  centre,  and  reciprocally  every  line  drawn  through 
the  centre  is  a  diameter. 

297.  If  both  of  the  axes  x',  y',  be  diameters,  the  trans 
formed  equation  will  not  contain  the  uneven  powers  of  either 
of  the  variables.     For  equations    (1),  (2),  and  (3)  must  in 
this  case  exist. 

298.  This  condition  is  always  fulfilled  in  curves  of  the 
second  order,  when  the  origin  of  the  co-ordinate  axes  is  taken 
at  the  centre,  and  their  direction  satisfies  equation  (1).    For, 
in  this  case,  the  first  powers  of  x'  and  y'  having  disappeared, 
as  well  as  the  term  in  x'y',  the  equation  will  contain  only  the 
square  powers  of  the  variables.     These  systems  of  diameters 
are  called  Conjugate  Diameters.    But  the  condition  of  passing 
through  the  centre  really  limits  this  property  to  the  Ellipse 
and  Hyperbola,  the  only  cases  in  which  equation  (4)  can  be 
satisfied  for  finite  values  of  a  and  b. 

299.  When  the  transformed  equation  contains  only  even 
powers  of  the  variables,  it  is  evident  that  if  this  equation  be 
satisfied  by  the  values   +  x',  -f  y',  it  will  also  be  for  — x', 
-f  y1 ;  — x',  — y' ;    +  x',  — y'  ;    that  is,  in  the  four  angles 
of  the  co-ordinate  axes,  there  will  be  a  point  whose  co-ordi 
nates  will  only  vary  in  signs.     If  the  axes  be  rectangular, 
the  form  of  the  curve  will  be  identically  the  same  in  each  of 
these  angles.     In  this  case,  it  is  said  to  be  symmetrical  with 
respect  to  the  axes.     In  the  ellipse  and  hyperbola,  for  ex 
ample,  these  curves  are  symmetrically  situated,  when   the 
co  ordinate  axes  coincide  with  the  axes  of  the  curves.    When 
a?'  and  y'  are  at  right  angles,  we  have  a  —  a  +  90°,  and  elimi 
nating  a!  from  equation  (1),  we  have 


CHAP.  V.]  ANALYTICAL  GEOMETRY.  203 

—  2C  sin  a  cos  a  +  B  (cos  2a  —  sin  2«)  -f-  2A  sin  a  cos  a  =  o, 

and 

(A  —  G)  tang  2a  +  B  =  o, 

an  equation  which  will  always  give  a  real  value  for  tang2«, 
from  which  we  deduce  two  real  values  for  tang  a.  For 

2  tang  a 

tang  2a  =  r ^—j- , 

1  —  tang2a 

hence, 

and 

2  (A  —  C)tanga  =  —  B  +  Btang2a, 

from  which  we  get 

tang2a ^— ~ tang  a  =  1. 

This  equation  will  make  known  the  two  values  of  a. 

Bat  the  product  of  the  roots  of  this  equation  being  equal 
to  the  second  member  taken  with  a  contrary  sign,  if  we  re 
present  these  roots  by  a  and  a',  we  shall  have 

/ | 

Hence  the  co-ordinate  axes  are  at  right  angles  (Art.  64),  and 
coincide  with  the  axes  of  the  curve. 

300.  We  may  readily  ascertain  whether  any  of  the  curves, 
represented  by  the  general  equation  we  have  been  discussing, 
have  asymptotes. 

For  this  purpose,  extracting  the  root  of  the  radical  part  of 
the  value  of  y,  we  have 

Bx  +  V^  VB2  —  4AC  BD  —  2AE 

"=         THT          1TA--*+   2AN/B^4AC 
K         K' 


C04  ANALYTICAL  GEOMETRY.  [CHAP.  V. 

Now,  it  is  obvious  that  as  x  increases,  all  the  terms,  in 
which  x  enters  as  a  part  of  the  denominator,  will  diminish, 
and  that  when  x  is  infinite,  the  value  of  y  will  reduce  to 


BD— 


j—  2AEv 
2  —  4AC/- 


~2A  2A  r        B2  —  4AC 

This  equation  represents  two  straight  lines,  to  which  the 
curve  continually  approaches  as  x  increases.  They  are 
therefore  the  asymptotes. 

As  this  equation  can  only  give  two  real  lines  when  B2  — 
4AC  ^>  o,  we  conclude  that  the  asymptotes  are  found  only 
in  the  third  class  of  curves. 

301.  Let  us  take  the  equation 


since  B2  —  4AC  >  o,  the  curve  belongs  to  the  third  class, 
corresponding  to  the  hyperbola. 
To  determine  its  asymptotes,  find  the  value  of  y.   We  obtain 


y  =  x  +  1  =fc  V  4*2  —  5x  +  2, 


«=  x 


Hence  the  equation  of  the  asymptotes  is 

y  =  x  +  I±2x  —  f. 

Constructing  this  equation,  we  can  determine  the  position 
of  the  asymptotes.  The  asymptotes  being  known,  if  we  de 
termine  the  point  in  which  the  curve  cuts  the  axis  of  x  or  yy 
we  may  construct  any  number  of  points  of  the  curve  by  the 
method  pursued  in  Art.  256. 


CHAP.  V.I  ANALYTICAL  GEOMETRY.  205 

EXAMPLES. 

1.  Find  the  asymptotes  of  the  curve  represented  by  the 
equation 

xy  —  2y  +  x  —  1  =  o. 

2.  Find  the  asymptotes  of  the  curve  represented  by  the 
equation 


3.  Find  the  asymptotes  of  the  curve  represented  by  the 
equation 

y2  —  2s2  —  2z/  +  Qx  —  3  =  o. 

4.  Find  the  asymptotes  of  the  curve  represented  by  the 
equation 

2  —  2x   —  -  x3  —  2x  —  2  =  o. 


Identity  of  Curves  of  the  Second  Degree  with  the  Conic 
Sections. 

302.  The  curves  which  have  been  discovered  in  the  dis 
cussion  of  the  general  equation  of  the  second  degree,  have 
presented  a  striking  analogy  to  the  Conic  Sections.     We  will 
resume  this  equation,  and  see  how  far  this  analogy  extends. 

303.  We  will  suppose  the  equation  to  contain  the  second 
power  of  at  least  one  of  the  variables,  and  that  the  system 
of  axes  is  rectangular.    We  have  found  for  the  general  value 
of  y  (Art.  279), 

J^ 

y~     ~2A  _____ 

±i 

18 


206  ANALYTICAL  GEOMETRY. 

The  expression 

1 


[CHAP.  V. 


is  the  equation  of  the  diameter  of  the  curve,  and  the  radical 
expresses  the  ordinate  of  the  curve  counted  from  this  diam 
eter.     Let  us  construct  these  re 
sults.      The     diameter    cuts    the 
axis  of  y  at  a  distance   from  the 

origin  equal  to  — ~-r->  and  makes 

an  angle  with  the  axis  of  x,  the 
trigonometrical  tangent  of  which 

T> 

is  —  g-r  •      Laying   off  a  length 

AD  =  —  Q-T  >  and  through  D  draw 
»ng  LDX',  making  the  angle  LOX  equal  to  that  whose  tan 
gent  is  —  wjj  >   LDX'  will    be    the  diameter  of  the  curve. 

Let  us  now  consider  any  point  M  whose  abscissa  AP  =  x, 
and  ordinate  PM  =  y.  Produce  PM  until  it  meets  the  di 
ameter  OX',  the  distance  PP'  will  represent  —  ^  (Ex  +  D) 

and  PM  the  radical  part  of  the  value  of  y.  But  as  the  equa 
tion  of  a  curve  is  simplified  by  referring  it  to  its  diameter, 
let  us  refer  the  curve  to  new  co-ordinates,  of  which  DP'  =  a: 
and  P'M  =  y',  and  call  the  angle  LOX,  a,  we  have 

x  =  —  x'  cos  a,  y  =  —  ^-r  (Ex  +  D)  +  y'. 

Substituting  these  expressions  in  the  general  value  of  y 
we  get 


CHAP.  V.]  ANALYTICAL  GEOMETRY. 

</'  ^ 

1        /(B2— 4  AC)  cos  W2—  2(BD— 2AE)cos  cu-'  +  D2—  4AF, 
2A  V 

or,  squaring  both  members, 

4A2*/'2  =  (B2  —  4AC)  cos  2a .  x'2  —  2  (BD  —  2AE) 
cos  a .  x'  4-  D2  —  4AF,        (2) 


or 


(BD  —  2AE)2 

Adding-^—    ,  W.x2        2     to  the  quantity  within  the  paren- 
c  (o  —  4AU)   COS   a 

theses,  and  subtracting  without  the  parentheses  its  equivalent 

(BD  —  2AE)2 
(B2  —  4  AC)  cos  «  7b2  _  A  A  Qy  -  2^'  tne  equation   becomes 

C  BD  —  2AE      7  2 

4Ay  =  (B2  -  4AC)  cos  ««     ,  -     2 


B2  —  4AC 
Let  us  introduce  for  x'  a  new  variable  a?",  such  that 

BD  —  2AE 
~(B2—  4  AC)  cos  a  ~ 

which  is  the  same  thing  as  transferring  the  origin  of  co-ordi- 

pr~\  _  o  A  "P 

nates  from  the  point  D  to  D',  so  that  DD'  =        lI 


The  equation  in  y'  and  x"  becomes 

4A'j/'2=  (B'-4AC)  cosV-o;"2-  -^  +  D2-4AF.    (3) 

And    since    under  this  form  it  contains   only  the  square 


208  ANALYTICAL  GEOMETRY.  [CHAP.  V 

powers  of  the  variables,  and  a  constant  term,  we  see  that  it 
can  only  represent  an  ellipse  or  hyperbola,  referred  to  their 
centre  and  axes,  or  conjugate  diameters.  It  will  represent 
an  ellipse  if  B2  —  4AC  is  negative,  arid  the  hyperbola  if  it  is 
positive. 

304.  This    reduction    supposes    that  the  last   transforma 
tion  is  possible.      But  this  will  always  be  the  case,  unless 

vw TTTTx '    which    represents    DD',    become    infinite. 

B2  —  4AC)  cos  a  J 

\vhich  can  only  be  the  case  when  (B2  —  4AC)  cos  a  =  o. 
But  cos  a  cannot  be  zero,  for  then  we  should  have  a  =  90°, 
which  would  make  A  =  o,  and  the  diameter  DX'  parallel  to 
the  primitive  axis  of  y,  a  case  which  we  excluded  at  first ; 
hence,  in  order  that  DD'  =  infinity,  we  must  have  B2  —  4AC 
=  o,  and  this  reduces  the  transformed  equation  to 

4Ay2  =  _  2(BD  —  2AE)  cos  a .  x'  +  D2  —  4AF,  (4) 
which  is  the  equation  of  a  parabola  referred  to  its  diameter 
DX'.     Thus,  in    every  possible    case,    the    equation    of  the 
second  degree  between  two  indeterminates  can  only  repre 
sent  one  or  the  other  of  the  conic  sections. 

305.  All  the  particular  cases  which  the  conic  sections  pre 
sent  may  be  deduced  from  these  transformations.     For  ex 
ample,  if  in  equation  (4)  we  suppose   BD  —  2AE  =  o,  the 
term  in  x'  disappears,  and  the  parabola  is  changed  into  two 
straight  lines  parallel  to  the  axis  of  x'.     If  D2  —  4AF  =  o 
also,  the  equation  will  represent  but  one  straight  line,  which 
coincides  with  this  axis.     If  in  equation  (3),  we  make  diffe 
rent  suppositions  upon  the  quantities  A,  B,  C,  D,  and  E,  we 
may  deduce  all   the  known  varieties  of  the  sections  which 
this  equation  represents,  which  proves  the  perfect  identity 
of  every  curve  of  the  second  order  with  the  conic  sections. 


OHAP.  V.]  ANALYTICAL  GEOMETRY.  209 

Tangent  and  Polar  Lines  to  Conic  Sections. 

306.  We  might  find  the  general  equation  of  a  tangent  line 
to  curves  of  the  second  order  by  following  the  same  process 
we  pursued  in  discussing  these  curves  in  detail.     But  as  the 
necessary  elimination  would  be  rather  long,  we  shall  here  make 
use  of  polar  co-ordinates  to  effect  the  desired  solution,  tlius : 
Refer  the  curve  to  polar  co-ordinates,  the  pole  being  on  the 
curve,  and  then  find  the  equation  of  condition  that  both  values 
of  the  radius  vector  become  zero,  when  it  will,  of  course,  be 
tangent  to  the  curve.     This  equation  of  condition  will  enable 
us  to  determine  the  value  of  the  tangent  of  the  angle  made  by 
the  tangent  line  with  the  axis  of  x. 

307.  Take  the  general  equation,  A?/2  +  TZxy  -f  Cz2  +  %  + 

Ez  +  F  =  o (1),  and  transform  it  by  means  of  the 

formulas,  x  =  xri  -f  r  cos  v,  y  =  y"  +  r  sin  v;  where  x"9  y"9 
are  the  co-ordinates  of  the  pole.     Arranging  the  transformed 
equation  with  reference  to  r,  it  will  be  of  the  form,  Mr2  +  Nr 

4-  P  =  o (2).    In  order  that  the  pole  may  be  on  the  curve, 

we  must  have,  P  =  0,  and  then  (2)  becomes,  Mr2  -f  2s>  =  o. 
Now  in  order  that  the  values  of  r  derived  from  this  last  equa 
tion  may  each  be  equal  to  zero,  we  must  have,  N  =  o.     Form 
ing  the  value  of  N"  by  actual  substitution,  and  placing  it  equal 
to  zero,  we  have,  2A?/"  sin  v  +  B  (xrr  sin  v  +  yrf  cos  ?•)  -f 
2Cx"  cos  v  +  D  sin  v  -f  E  cos  v  ==  0,  which  gives,  tang  v  = 

By"  +  2Cz"  +  E 

9  1  "  4-  B  "  -4-  D'  ^°r  *^e  tangent  of  the  angle  made  by  the 

tangent  line  with  the  axis  of  x.     Therefore  the  equation  of 

By"  -f  ^Cx"  -f  E 

this  tangent  is,  y-y»  =  -~2L/"  +  Ex"  +  D  (*  "~  *") J  °r?  ^ 
reducing, 

18*  2B 


210  ANALYTICAL  GEOMETRY.  [Cuip.  V. 


+  Bz"  +  D)  y  +  (2Gr"  -4-  By"  +  E)  &  +  D#"  -f 
Ex"  +  2F  =  o  .......  (3) 

308.  Having  found  the  general  equation  of  the  tangent  line 
to  Conic  Sections,  we  are  now  prepared  to  demonstrate  a  re 
markable   and  beautiful  property  of  these  curves,  namely  ; 
That  if  from  any  point  in  the  plane  of  a  conic  section  we 
draw  any  number  of  secants,  and  at  the  points  of  intersection, 
of  the  curve  with  these  secants,  pairs  of  tangents  be  drawn  to 
the  curve,  then  the  points  of  intersection  of  these  pairs  of 
tangents  will  all  be  found  upon  a  straight  line;   and,  con 
versely,  If  we  take  any  right  line  in  the  plane  of  a  conic  sec 
tion,  and  from  every  point  of  this  line  draw  pairs  of  tangents 
to  the  curve,  and  connect  the  points  of  contact  of  each  pair 
ly  a  right  line,  all  these  last  lines  will  meet  in  a  common 
point.     Let  there  be  a  point  P  without  the  curve,  whose  co 
ordinates  are  xf,  yr,  and  let  it  be  proposed  to  draw  from  this 
point  a  tangent  to  the  curve.     The  question  is  then  reduced 
to  finding  the  point  of  contact,  and  as  this  point  is  upon  the 
curve,  we  must  have  the  equation, 
A?/"2  +  Bz'y  '  +  Vx"2  +  Dy"  +  Ex"  +  F  =  o  .....  (4) 

Because  the  point  P  is  upon  the  tangent  line,  we  must  have 
the  equation, 

(2A#"  +  Ba?"  +  D)  yr  +  (2Ca/'  +  B?/"  +  E)  x1  +  D#"  + 

Ez"  +  2F  =  o  .....  (5) 

The  combination  of  (4)  and  (5)  would  give  the  desired 
values  of  x"  and  y"  .  Instead  of  doing  this,  however,  we 
may  obtain  these  points  by  constructing  the  geometric  loci  of 
(4)  and  (5)  under  the  supposition  that  x"  and  y1'  are  variables. 
Under  this  hypothesis,  (4)  represents  the  given  curve,  and  (5) 
represents  a  right  line  two  of  whose  points  are  the  required 
points  of  contact,  and  therefore  it  must  be  the  equation  of 


CHAP.  V.]  AXALYTICAL  GEOMETRY.  211 

the  secant  connecting  those  points.  Now  if  this  last  line  be 
required  to  pass  through  a  point  0  whose  co-ordinates  are  a 
and  by  these  co-ordinates  must  satisfy  (5)  when  substituted  for 
x"  and  yn f,  and  it  then  becomes, 

(2A6  +  Btf  -f  D)  y'  +  (2Ca  +  B6  -f  E)  x'  +  Db  +  Ea  + 
2F  =  o (6) 

N^w  in  this  last  equation  the  co-ordinates  x',  yf,  belong  to 
a  point  P,  such  that  if  from  it  two  tangents  be  drawn  and 
their  points  of  contact  connected  by  a  line,  this  line  passes 
through  the  point  0  whose  co-ordinates  are  a  and  b.  Let  us 
now  suppose  the  point  P  to  change  its  position ;  it  is  evident 
that  of  all  the  positions  it  can  take,  there  is  an  infinite  num 
ber  such,  that  drawing  from  them  pairs  of  tangents  to  the 
curve,  and  connecting  the  points  of  contact  of  each  pair  by  a 
right  line,  all  these  last  lines  will  pass  through  the  point  0; 
and  all  such  positions  of  the  point  P,  and  none  others,  will  be 
given  by  those  values  of  x'  and  yr,  which  satisfy  (6).  Then, 
if  in  (6)  x1  and  y'  be  regarded  as  variables,  (6)  will  represent 
the  geometric  locus  of  these  positions  of  the  point  P.  Under 
this  supposition,  however,  (6)  represents  a  straight  line,  and 
hence  the  truth  of  the  first  branch  of  the  theorem. 

309.  Again,  if  any  line  L,  be  given  in  the  plane  of  a  conic 
section,  this  line  may  be  represented  by  (6),  and  then  the 
values  of  a  and  b  which  satisfy  (6)  without  x'  and  yr  ceasing 
to  be  indeterminate,  will  fix  a  point  0  having  with  the  line  L 
the  relation  enunciated  in  the  second  branch  of  the  proposi 
tion.     The  point  0  is  called  the  pole  of  the  line  L,  which  last 
line  is  called,  relatively  to  the  point  0,  the  polar  line.     This 
nomenclature  must  not,  however,  be  confounded  with  polar 
co-ordinates. 

310.  The  properties  of  Poles  and  Polar  Lines  are  extremely 


212 


ANALYTICAL  GEOMETRY. 


[CHAP.  V. 


valuable  in  many  graphic  constructions  relating  to  Conic  Sec 
tions,  but  the  limits  of  this  treatise  do  not  permit  a  full  inves 
tigation  of  them.  We  shall  therefore  confine  ourselves  to 
showing  how  the  Pole  may  be  found  when  we  know  the  Polar 
Line,  and  reciprocally ;  and  then  how  they  may  be  applied  to 
drawing  tangents  to  Conic  Sections. 

311.    First,  knowing  the  pole  0,  to   find   the   polar  line 
(Fig.  a).    From  the  pole  0  draw  any  two  secants  as,  OB,  OA ; 


then  draw  CD  and  AB,  forming  the  incribed  quadrilateral 
ABDC.  The  intersection  of  the  sides  AB  and  CD  gives  one 
point  P  on  the  polar  line,  and  the  point  H,  where  its  diagonals 
BC  and  AD  meet,  is  another  point,  so  that  PH  is  the  polar 
line  for  the  pole  0.  Had  H  been  the  given  pole,  situated 
within  the  curve,  then  by  drawing  through  it  any  two  secants, 
as  AD  and  BC,  and  connecting  the  points  A,  B,  D,  C,  where 
they  intersect  the  curve,  so  as  to  form  the  inscribed  quadri- 


CHAP.  V.]  ANALYTICAL  GEOMETRY.  213 

lateral  ABDC,  the  intersection  of  its  sides  prolonged,  -would 
have  fixed  the  points  P  and  0,  and  PO  would  have  been  the 
polar  line  for  the  pole  H.* 

312.  Let  it  now  be  required  to  draw  a  tangent  to  the  Conic 
Section  ATT',  from  the  point  P  without  the  curve.     From  P 
draw  any  two  lines  PA,  PC,  cutting  the  curve  at  A,  B,  D,  C. 
Then  draw  BD  and  AC,  and  prolong  them  till  they  meet  at 
0.     There  will  thus  be  formed  the  quadrilateral  ABDC,  in 
scribed  within  the  curve.     Draw  its  diagonals  AD  and  BC, 
meeting  at  H.     Join  0  and  H  by  the  right  line  OH,  which 
will  cut  the  curve  at  the  two  points  T  and  T'.     These  will  be 
the  points  of  contact,  and  by  joining  them  with  P  we  shall 
obtain  the  required  tangents  PT,  PT'. 

313.  In  the  second  case,  suppose  the  given  point  P  (Fig.  b) 


T 

to  lie  upon  the  curve.     Assume  any  three  other  points  as,  A, 
B,  D,  upon  the  curve.     Draw  DP,  and  AB,  intersecting  at 

*  See  note  at  end  of  this  subject. 


214  ANALYTICAL  GEOMETRY.  [CHAP.  V. 

M ;  also  draw  BP  intersecting  AD  prolonged,  at  R ;  and  then 
draw  RM.  Now  change  one  of  the  three  assumed  points,  as 
B,  to  any  other  position,  as  C,  an$  go  through  the  same  con 
struction  ;  that  is,  draw  AC  meeting  DP  at  S;  then  draw 
CP  meeting  AD  prolonged,  at  N;  and  then  draw  NS,  and 
prolong  it  until  it  meets  RM  at  T,  which  will  be  a  point  of  the 
tangent,  and  drawing  TP,  it  will  be  the  tangent  line  required. 
A  line  from  T  to  A  would  also  be  tangent  to  the  curve  at  A- 

314.  The  student  will  find  it  a  valuable  exercise  to  examine 
and  discuss  poles  and  polar  lines  for  each  of  the  varieties  of 
Conic  Sections  separately.  And  we  may  here  mention  that  in 
the  case  of  the  Parabola,  he  will  find  the  directrix  to  be  the 
polar  line  of  the  focus,  and  reciprocally,  the  focus  to  be  the 
pole  of  the  directrix.  Hence,  if  any  chord  be  drawn  through 
the  focus  of  a  parabola  and  two  tangents  be  drawn  at  its  ex 
tremities,  these  tangents  will  intersect  on  the  directrix.  It 
will  also  be  found  that  these  tangents  are  perpendicular  to  each 
other. 

315.  Note. — The  construction  of  Art.  311  presents  one  of  those  instances 
in  which  a  resort  to  the  ordinary  analytic  methods,  as  a  means  of  proof,  would 
he  attended  with  much  disadvantage,  on  account  of  the  elimination  required. 
The  most  convenient  and  direct  demonstration  reposes  upon  the  theory  of 
Harmonic  pencils,  with  which  we  cannot  suppose  the  pupil  familiar,  as  it  has 
not  yet  found  its  way  into  our  geometries.  We  may  mention,  however,  for 
the  benefit  of  the  student  acquainted  with  the  principles  of  Linear  Perspective^ 
that  a  very  simple  and  elegant  proof  may  be  established  by  its  means :  de 
pending  on  the  fact  that  pairs  of  secants  uniting  the  corresponding  extremities 
of  parallel  chords  of  a  conic  section,  meet  on  the  diameter  bisecting  these 
chords.  The  constructions  of  Arts.  312,  313,  are  immediate  consequences  of 
that  of  Art.  311. 


CHAP.  V.j  ANALYTICAL  GEOMETHY.  215 

Intersection  of  Curves. 

316.  Before  closing  this  Qiscussion,  we  will  show  how  the 
principles  developed  in  Art.  92  may  be  applied  to  determine 
the  points  of  intersection  of  two  curves. 

If  the  curves  intersect,  the  co-ordinates  of  the  points  of 
intersection  must  satisfy  the  equations  of  both  curves.  These 
equations  must  therefore  have  common  roots,  and  the  deter 
mination  of  these  roots  will  make  known  the  co-ordinates  of 
the  points  of  intersection. 

317.  Take  the  equations 

y  =  ~x,     y2  +  ay  —  x*  +  bx. 

Determining  the  values  of  x  and  y  by  elimination,  we  find 

x  =  o,  y  =  o  ;     x  =  —  b,  y  =  —  a. 

Hence  the  straight  line  meets  the  curve  in  two  points, 
which  may  be  constructed  from  the  values  which  have  been 
found  for  the  co-ordinates. 

318.  Let  us  take  the  equation 


if  — 

Subtracting  the  first  equation  from  the  second,  we  have 
for  the  first  equation 

2y  =  o, 

which  gives  y  =  o. 

Substituting  this  value  in  either  of  the  given  equations, 

we  find 

x  =  o,  and  x  =  1. 

The  curves  therefore  intersect  in  two  points. 


216  ANALYTICAL  GEOMETRY.  [CHAP.  V. 

319.  Let  us  take  for  another  example, 

y2  —  2xy  +  x2  —  %  — -1  =  o, 
y2  —  %xy  +  x*  +  x  —  o. 

Determining  the  first  equation  in  x  by  means  of  the  greatest 
common  divisor,  we  find 

9x2  +  Wx  +  1  =  o, 
which  gives  for  the  values  of  x, 

x  =  —  1,  and  x  =  — \. 

Substituting  these  values  in  the  last  divisor  placed  equal  to 

zero,  we  have 

y  =  o,  y  =  —  j. 

The  given  curves  have  therefore  two  points  of  intersec 
tion,  which  may  be  constructed  by  methods  previously  ex 
plained. 

320.  As  two  equations,  one  of  the  mth,  and  the  other  of 
the  nth  degree,  may  have  a  final  equation  of  the  mnth  degree; 
it  follows  that  the  curves  represented  by  these  equations  may 
intersect  each  other  in  mn  points.     As  the  roots  of  a  final 
equation,  the  degree  of  which  exceeds  the  2d,  are  not  readily 
constructed,  a  method  is  often  used,  which  consists  in  draw 
ing  a  line  which  shall  be  the  locus  of  all  the  points  of  inter 
section,  and  thus   their  situation  will  be  determined. 

321.  To  explain  this  method.     Let 

y=f(x)     y  =  <p(ff)* 


*  A  quantity  is  said  to  be  a  function  of  another  quantity,  when  it  depends 
upon  it  for  its  value.  The  expressions  f(x\  <f>  (#),  &c.,  are  used  to  denote 
any  functions  of  x,  aud  are  read, /function  of  x,  $  function  of  x,  &c. 


CHAP  V.]  ANALYTICAL  GEOMETRY.  217 

be  the  equations  of  two  curves.  If  they  intersect,  the  co 
ordinates  x  and  y'  of  their  intersection  must  satisfy  these 
equations,  and  we  have 

»'=/(*')   V  =  ?(*');     (i) 

adding  these  equations  together,  and  then  multiplying  them 
by  each  other,  we  have 

2y- =/(*')  + 9  (*%        (2) 
/=/(*')  *?(*')•        (3) 

Now,  either  of  the  equations  (2)  or  (3)  gives  a  true  relation 
between  the  co-ordinates  x',  y ',  of  the  points  of  intersection; 
and  by  supposing  x  and  y  to  vary,  this  equation  will  express 
the  relations  between  the  co-ordinates  of  a  line,  one  of  whose 
points  will  be  the  required  line  of  intersection 

It  may  be  remarked,  that  in  combining  the  given  equations 
we  should  endeavour  to  lead  to  equations  which  are  most 
readily  constructed;  the  straight  line  and  circle  being  pre 
ferred  to  any  other. 

EXAMPLE. 

From  a  given  point  without  an  ellipse,  draw  a  tangent  tc 
the  curve. 

We  have  for  the  equation  of  the  ellipse. 

A2*/2  +  B2*2  =  A2B2,     (1) 
and  for  that  of  the  tangent, 

A*yy"  +  Wxx"  =  A2B2. 

19  2c 


218 


ANALYTICAL  GEOMETRY. 


[CHAP.  V. 


Let  x',  y,  be  the  co-ordinates  of  the  given  point  Q,  they 
must  satisfy  the  equation  of  the  tangent,  and  we  have 

A?y'ij"  +  BVa?"  =  A2B2.     (2) 

From  the  equations  (1)  and  (2)  we  can  readily  find  the 
values  of  x"  and  y",  and  thus  determine  P. 


Now,  equation  (2)  is  not  the  equation  of  any  straight  line, 
but  only  gives  the  relation  between  CM  and  MP.  But  if  we 
suppose  x"  and  y"  to  vary,  this  equation  will  express  the 
relation  between  a  series  of  points,  one  of  which  will  be  P ; 
and  therefore  if  the  line  it  represents  be  constructed,  it  will 
pass  through  P,  and  its  intersection  with  the  given  ellipse 
will  make  known  the  point  P.  Constructing  the  line  whose 
equation  is 

A*y'y"  +  BVa?"  =  A2B2, 

we  find  it  to  be  BPP',  and  that  it  intersects  the  ellipse  in  two 
points.  Two  tangents  can  therefore  be  drawn  to  the  curve, 
QP,  and  QP'. 


CHAP.  YL]  AXALYTICAL  GEOMETRY.  219 


CHAPTER  YL 

CURVES   OF   THE    HIGHER   ORDERS. 

322.  HAVING  completed  the  discussion  of  lines  of  the  second 
order,  we  might  naturally  be  expected  to  proceed  to  an  inves 
tigation  of  those  of  the  higher  orders ;  but  the  bare  mention 
of  the  number  of  those  in  the  next,  or  third  order  (for  they 
amount  to  eighty),  is  quite  sufficient  to  show  that  their  complete 
discussion  would  far  exceed  the  limits  of  an  elementary  trea 
tise  like  the  present.  Nor  is  such  an  investigation  necessary; 
we  have  examined  the  Conic  Sections  at  great  length,  because, 
from  their  connexion  with  the  system  of  the  world,  every  pro- 
perty  of  these  curves  may  be  useful ;  but  it  is  not  so  with 
curves  of  the  higher  orders ;  generally  speaking,  they  possess 
but  few  important  properties,  and  may  be  considered  more  as 
objects  of  mathematical  curiosity  than  of  practical  utility. 
The  third  order  is  chiefly  remarkable  from  its  examination 
having  been  undertaken  by  Newton.  Of  the  eighty  species 
now  known,  seventy-two  were  discussed  by  him.  and  eight 
others  have  since  been  discovered.  The  varieties  of  the  next, 
or  fourth  order,  are  thought  to  number  several  thousands.  A 
systematic  examination  of  curves  being  thus  impossible,  all 
we  can  do  is  to  give  a  selection,  confining  our  attention  princi 
pally  to  such  as' may  merit  special  notice,  either  on  account 
of  their  history,  or  for  the  possession  of  some  remarkable  me 
chanical  property.  Others  we  shall  notice  in  order  that  the 
student  may  not  be  entirely  unfamiliar  with  them  when  he 


220  ANALYTICAL  GEOMETRY.  [CHAP.  VI 

may  meet  with  some  allusion  to  them  in  the  higher  brandies 
of  analysis.  And  as  this  matter  of  tracing  the  geometrical 
form  and  figure  of  a  curve  from  its  equation,  is  one  of  surpass 
ing  importance  in  the  practical  application  of  mathematics,  we 
shall  commence  by  selecting  an  example  well  calculated  to 
exhibit  a  further  illustration  of  those  principles  by  which  we 
have  already  discussed  the  Conic  Sections,  as  well  as  to  show 
clearly  the  general  method  of  procedure  in  such  cases. 
323.  We  begin  then  with 

The  Lemniscate  Curve, 
represented  by  the  equation, 

y*  —  96«y  +  100  A2  —  x4  =  o (A). 

Here  let  us  observe  that,  in  the  discussion  of  any  curve,  the 
sole  difficulty  consists  in  resolving  the  equation  by  which  it  is 
defined.  If  this  obstacle  can  be  overcome,  we  may  readily 
trace  its  course.  For,  suppose  that  the  equation  of  the  curve 
has  been  solved,  and  that  X,  X',  X",  etc.,  represent  the  roots 
of  ?/,  these  roots  being  functions  of  x;  the  question  is  at  once 
reduced  to  an  examination  of  the  particular  curves,  which  are 
represented  by  the  separate  equations, 

y-X,        y-X',        y  =  X",etc. 

This  examination  will  be  effected  by  giving  to  x  every  pos 
sible  value,  as  well  negative  as  positive,  which  the  functions 
X,  X',  Xr/,  etc.,  admit  of,  without  becoming  imaginary;  and 
the  resulting  curves  will  be  the  different  branches  of  the  curve 
denoted  by  the  given  equation.  The  extent  and  direction  of 
these  branches  will  depend  upon  the  different  solutions  which 
correspond  to  their  particular  equations.  If  any  of  the  equa 
tions  y  =  X,  y  =  X',  etc.,  exist  for  infinite  values  of  x9  it  fol- 


CHAP.  VI.]  ANALYTICAL  GEOMETRY.  2*1 

lows  that  the  corresponding  branches  extend  indefinitely  in  the 
direction  of  these  values. 

324.  The  present  example  offers  no  difficulty  in  the  solution 
of  its  equation,  which,  being  effected  by  the  method  for  qua 
dratic  equations,  gives  us, 


y  =  ±  V  48a2  ±  v/2304a4  —  lOOa2^  +  2:* (B), 

or  putting,  2304a4  —  lOOaV  +  z4  =  1ST,  the  four  values  of  y 
become, 

y  =  N/48a2  +  x/N (1), 


y=\/48a2— VN (2), 

y  =  —  V  48a2  +  v/N (3), 


(4), 

It  is  now  required  to  ascertain  each  of  the  curves  which 
C0v;se  equations  represent.  We  see,  in  the  first  place,  that  the 
values  (3)  and  (4)  differ  from  (1)  and  (2)  only  in  the  sign,  and 
consequently  must  represent  branches  similar  to  those  repre 
sented  by  (1)  and  (2),  but  differently  situated  with  reference  to 
the  axis  of  x.  Further,  as  the  quantity  of  N  contains  only 
even  powers  of  x,  its  value  will  not  be  changed  by  substituting 
a  negative  for  a  positive  value  of  x.  The  parts  of  the  curve 
which  lie  on  the  right  of  the  axis  of  y,  are,  then,  similar  to 
those  situated  on  the  left  of  this  axis.  Hence  the  curve  is 
divided  by  the  co-ordinate  axis  into  four  equal  and  svmmetri- 
cal  parts.  Let  us  now  proceed  to  a  more  minute  examination 
of  the  values  (1)  and  (2),  beginning  with  (1).  This  value  of 
y  can  only  be  real  so  long  as  N  is  positive,  and  we  know  from 
19* 


222 


ANALYTICAL  GEOMETRY. 


[CHAP.  VI. 


algebra  that  in  an  expression  of  this  kind  a  change  of  sign 
can  only  occur  by  its  passing  through  zero,  and  therefore  we 
can  find  the  limits  to  the  real  values  of  y  by  writing  N  =  x4 — 
lOOaV2  -j-  2304a4  =  o,  which  equation  gives  by  its  solution, 
x  =  ±  6a,  and  x  =  ±  8#,  and  hence  (1)  may  be  written, 


48a2  -f  V(x  —  Qa)  (x  +  6a)  (x  —  8a)  (x  +  8a) (5). 


In  this  equation,  x  =  o  gives  y  =  \/96a2  for  the  point  C 
(Fig.  1),  in  which  the  curve  cuts  the  axis  of  y.     Between  the 


f  G 

limits  x  —  o  and  x  —  6a,  N  is  positive  and  y  is  real,  and  as  x 
increases  from  o  to  6a,  y  diminishes  from  \/96a2  to  \/48a2, 
which  last  value  corresponds  to  the  point  D,  at  which  a  line 
parallel  to  the  axis  of  y  is  tangent  to  the  curve.  For  values 
of  x  greater  than  Qa  and  less  than  8a,  the  factor  (x  —  So) 
alone  becomes  negative,  and  consequently  renders  y  imaginary, 
so  that  no  portion  of  the  curve  is  found  between  the  parallels, 
FD  and  GE,  to  the  axis  of  y  at  distances  AF  and  AG,  from 
the  origin  equal  respectively  to  Qa  and  8a.  For  x  =  8 or,  we 
get  y  =  v/48a2,  giving  the  point  E,  at  which  EG  parallel  to 
the  axis  of  y  is  tangent  to  the  curve.  All  values  of  x  greater 
than  Sa  render  N,  and  consequently  y,  positive ;  hence,  from 
E  the  curve  extends  indefinitely  in  the  direction  EH.  Similar 
branches  will  be  found  on  the  left  of  the  axis  of  «/,  by  attri- 


CHAP.  VI.]  ANALYTICAL  GEOMETRY. 

buting  negative  values  to  rr,  so  that  equation  (1)  represents  the 
portions  of  the  curve  exhibited  in  Fig.  1.  If  in  the  general 
equation  (A),  we  make  y=  v96a2,  we  obtain,  x2  =  o,  and 
x  =  dz  lOfl.  The  first  gives  x  =  db  0,  which  shows  that  at  the 
point  C,  the  parallel  I'd  to  the  axis  of  x,  is  tangent  to  the 
curve,  while  the  other  two  values  of  x,  viz.  ±  10a,  give  the 
points  I  and  I7  at  which  the  parallel  cuts  the  two  indefinite 
branches.  Now  let  us  examine  (2).  By  a  transformation 
similar  to  that  used  in  the  discussion  of  (1),  this  second  value 
of  y  may  be  written, 


\ 


/48a2—  Viz  —  6 


V  (x  —  6a)  (x  +  6a)  (x  —  Sa)  (x  +  Sd) (6). 


G'F' 


F GK 


In  this  equation  x  =  o  gives  y  =  t>,  which  shows  that  the 
curve  passes  through  the  origin.    As  x  increases  from  zero  up 
to  6c7,  y  increases  from  zero  to  N/48a2,  which  last  value  gives 
the  point  D  (Fig.  2),  at  which 
this  branch   joins   that   of    CD 
(Fig.  1),  and  both  have  a  com 
mon  tangent,  DF,  parallel  to  the 
axis  of  y.     For  all  values  of  x 
greater  than  6tf,  but  less  than  Sa, 
the  factor  (x  —  Sa)  alone  becomes  negative,  rendering  N  nega 
tive,  and  consequently  y  imaginary,  so  that  no  part  of  the 
curve  represented  by  equation  (6)  is  found  between  the  two 
lines  DF  and  EG  drawn  parallel  to  the  axis  of  y.  and  at  dis 
tances  AF  and  AG  from  the  origin  equal  respectively  to  Qa 
and  8 a.     For  x  =  Sa,  (6)  gives  y  =  v/48a2,  for  the  point  E, 
in  which  the  branch  EK  joins  the  branch  EH  (Fig.  1),  and 
both  have  the  common  tangent  EH  parallel  to  the  axis  of  y. 
From  the  form,  of  equation  (2),  it  is  apparent  that  a  negative 


224  ANALYTICAL  GEOMETRY.  [CHAP.  VI. 

value  for  N  is  not  the  only  circumstance  which  will  render  y 
imaginary.  For  y  is  plainly  imaginary  whenever  x  has  such 
a  value  as  to  render  </N>48a2.  We  then  obtain  the  limits 


by  writing,  v/N  =  v/^^lOOaV  -f  2304a4  =  48a2,  which 
equation  when  resolved  gives,  x2  =  o  and  x  —  rb  10a.  The 
first  of  these  values  of  x  corresponds  to  the  origin.  The  other 
two,  ±  lOa,  give  the  points  K  and  K'  at  which  the  branches 
EK  and  E'K'  are  cut  by  the  axis  of  x.  Thus,  for  all  values 
of  x  between  the  limits  x  =  80,  and  x  =  10<z,  equation  (6) 
gives  real  values  for  y,  and  for  all  values  of  x  greater  than 
10a  y  is  imaginary,  so  that  the  branches  represented  by  (6) 
are  limited  at  K  and  K'  by  parallels  to  the  axis  of  y.  More 
over,  as  x  increases  from  Sa  to  100,  y  diminishes  from  v/48a2 
to  zero,  so  that  between  the  points  E  and  K  the  branch  EK 
has  the  form  represented  in  the  diagram.  Again,  if  in  the 
general  equation  (A)  we  make  x  —  100,  we  obtain,  y2  —  o, 
y  =  V96a2.  The  first  gives  y  =  =b  o,  and  shows  that  at  K 
and  K'  the  parallels  to  the  axis  of  y  are  tangent  to  the  curve  ; 
the  other  value,  \/  96a2,  corresponds  to  the  points  I  and  I' 
(Fig.  1).  By  giving  negative  values  to  rr,  we  find  similar 
branches  to  exist  on  the  left  of  the  axis  of  y,  so  that  the  por 

tions  of  the  curve  defined 
by  (2)  are  such  as  are  re 
presented  in  Fig.  2.  As 
we  have  already  remarked, 
equations  (3)  and  (4)  repre 
sent  equal  branches  situated 
below  the  axis  of  x.  In. 

Fig.  3  are  shown  the  branches  represented  by  (1)  and  (2),  and 
Fig.  4  exhibits  the  entire  curve. 


CHAP.  VI.] 


ANALYTICAL  GEOMETRY. 


•225 


Let  us  now  examine  if  this  curve  has  asymptotes.  By  ex 
tracting  the  square  root  of  the  quantity  N,  equation  (B)  may 
be  "written, 


etc.) 


or  taking  the  upper  sign  only, 


rt  ,  4900a6 

-  2a2-  -    --- 


Extracting  the  square  root  again,  we  have, 


etc. 


a2       99a*  x 

—  5—  fir  ......  etc.)  ......  (7).' 


226 


ANALYTICAL  GEOMETRY. 


[CHAP.  VI. 

Now  as  x  increases,  those  terms  in  this  equation  which  contain 
x  in  the  denominator  will  diminish,  and  when  x  =  oo,  they 
may  be  all  neglected  after  the  first;  equation  (7)  then  reduces 
to  y  =  db  x,  which  is  the  equation  of  two  rectilinear  asymp 
totes  to  the  curve,  passing  through  the  origin  and  making 
angles  of  45°  and  135°  with  the  axis  of  x.  By  combining 
the  equation  of  the  asymptote  with  that  of  the  curve,  we  find 
that  the  origin  is  the  only  point  in  which  they  intersect.  The 
asymptotes  are  represented  in  Fig.  4  by  the  lines  RAB/,  SAS'. 
The  polar  equation  of  this  curve  is  readily  found  to  be, 

r4  —  4a V)  cos  2$  —  2aV 


o. 


Its  discussion  is  left  as  an  exercise  for  the  student. 

E  325.  The  Cissoid  of  Diodes  (Fig. 
5).—  Let  ADBD'  be  a  circle  of  which 
AB  is  the  diameter  and  EBF  an  in 
definite  tangent  at  the  point  B  ;  draw 
from  A  any  line  AI,  cutting  the  cir- 
cumference  at  o  and  the  tangent  at 

I,  then  take  on  this  line  the  distance 
i? 

Am  =  01;  it  is  required  to  find  the 
B    locus  of  the  points  m,  mr,  etc.     Take 
A  as  the  origin  of  a  system  of  rec 
tangular  co-ordinates,  AB  being  the 
axis    of    x.      Then   put   AB  =  2a, 
An  =  z,  and  mn  =  y.     Now,  since 
Am  =  ol,  An  will  be  equal  to  pB, 
and  the  similar  triangles  Anm  and 
F  Apo  give,  An  :nm  :  :  Ap  :  po,  that  is, 


=       —  ,  and  y 


CHAP.  TL] 


ANALYTICAL  GEOMETRY. 


C27 


\/   —i — .     For  the  sake  of  convenience,  let  us  tabulate  tho 

V          () /*     <Y 


—  x 


corresponding  values  of  x  and  ?/,  thus 


123 

4 

5 

6 

Val.  x 

o              a       |    <  '2a 

2a 

>2a 



Yal.  y 

db  o         ±  a         real 

±00 

imag. 

imag. 

From  (1)  we  see  that  the  curve  passes  through  the  origin ; 
from  (2)  that  it  bisects  the  semicircular  arcs  ADB  and  AD'B 
at  the  points  D  and  D';  from  (3)  that  for  all  values  of  x  less 
than  2a  there  are  two  real  and  equal  values  for  y  with  contrary 
signs ;  from  (4)  that  there  is  an  infinite  ordinate  at  B,  or  that 
EBF  is  an  asymptote  to  the  curve.  From  (5)  we  perceive 
that  no  point  of  the  curve  lies  to  the  right  of  this  asymptote, 
and  from  (6)  that  no  part  of  it  is  found  to  the  left  of  A,  and 
as  the  curve  is  symmetrical  with  respect  to  the  axis  of  x,  its 
form  is  such  as  represented  in  the  diagram.  This  curve  was 
invented  by  Diocles,  a  mathematician  of  the  third  century, 
and  called  by  him  the  Cissoid,  from  a  Greek  word  signifying 
"  ivy,"  because  he  fancied  that  the  curve  climbs  up  its  asymp 
tote  as  ivy  does  up  a  tree.  He  employed  it  in  solving  the 
celebrated  problem  of  inserting  two  mean  proportionals  be 
tween  given  extremes. 

326.  The  Conchoid  of  Nicomedes  (Fig.  7).— Let  BX  be  an 
indefinite  right  line,  A  a  given  point,  from  which  draw  ABC 
perpendicular  to  BX,  and  also  draw  any  number  of  straight 
lines  Aom,  Ao'm",  etc.;  upon  each  of  these  lines  take  om 
and  omr,  o'm"  and  o'rn'",  each  equal  to  BC,  then  the  locus 
of  these  points  m,  m',  m",  mf/r,  etc.,  is  the  conchoid.  The 


228 


ANALYTICAL  GEOMETRY. 


[CHAP.  VI. 


branch  HCG  is  called  the  superior  conchoid,  and  the  other 
portion,  FADAE,  the  inferior  conchoid:  both  conchoids  form 
but  one  curve,  that  is,  are  both  defined  by  the  same  equation. 


Fij.7. 


BC  is  called  the  modulus,  and  BX  the  base  or  rule.  Let  us 
now  find  the  equation  of  the  curve  from  its  mode  of  genera 
tion.  The  curve  may  be  regarded  as  the  locus  of  the  points 
of  intersection  of  the  lines  mm',  Am",  etc.,  with  the  circles 
which  have  their  centres  at  0,  0',  etc.,  and  their  radii  each 
equal  to  BC.  The  equation  of  one  of  these  circles  would  be, 

(x  —  x')2  -f  y1  =  b* (1),   and  that  of  one  of  the  lines 

Am  is,  y  -f  a  —  dx (2).     Now  the  centre  of  this  circle 

must  be  at  the  point  in  which  Am  cuts  the  axis  of  x,  which 

gives,  x'  =  -j.     Hence  (1)  becomes, 

f  y2  =  b2 (3). 


Now  to  get  the  desired  locus,  we  must  eliminate  d  between  (2) 
and  (3),  in  terms  of  general  co-ordinates,  and  we  thus  obtain, 


or, 


CHAP.  VI.] 


ANALYTICAL  GEOMETRY. 


for  the  equation  of  the  curve,  which  we  now  proceed  to  dis 
cuss,  observing  that  we  may  distinguish  the  cases  according  as 
we  have,  b  >  a,  b  =  a,  or  5  <  a. 
327.  CASE  I.    6a. 


1 

2 

3 

4 

5 

6 

7 

Q 

VaLy 

0 

6 

<b 

>& 

—  a 

—  b 

<-<• 

>-a,<-J 

Val.2? 

00 

0 

real 

imag. 

0 

0 

real 

real 

From  (1)  XX'  is  an  asymptote ;  from  (2)  the  curve  passes 
through  C ;  from  (3)  and  (4)  the  curve  extends  from  the  base 
upwards  to  C,  and  no  higher;  hence  the  branch  HCG.  Again, 
from  (5)  and  (6)  the  curve  passes  through  A  and  D  if  BD  =  b; 
from  (7)  there  is  an  indefinite  branch  AE,  to  which  the  base 
is  an  asymptote ;  and  from  (8)  the  curve  exists  between  A  and 
D,  and  since  the  curve  is  symmetrical  with  reference  to  the 
axis  of  y,  its  form  is  as  represented  in  the  diagram. 

328.  CASE  II.  b  =  a.  The  loop  Am'DA  disappears  by  the 
coincidence  of  the  points  A  and  D ;  otherwise  the  curve  is  of 
the  same  form  as  in  the  first  case. 

CASE  III.  b  <C  CL  In  this  case  the  superior  conchoid  is  not 
altered,  but  the  inferior  conchoid  becomes  a  curve  similar  to 
it,  the  point  D  falling  between  A  and  B.  The  point  A  be 
comes  what  is  known  as  a  conjugate  or  isolated  point,  that  is, 
a  point  whose  co-ordinates  satisfy  the  equation  of  the  curve, 
and  which  is  therefore  a  point  of  the  curve,  but  is  entirely 
isolated  or  disconnected  from  the  branches  of  the  curve.  The 
generation  of  the  conchoid  affords  a  good  example  of  the 
20 


230  ANALYTICAL  GEOMETRY.  [CHAP.  VI. 

nature  of  an  asymptote,  for  the  distances  om,  ofmff,  etc.,  must 
always  remain  each  equal  to  BC,  and  this  plainly  causes  the 
curve  to  approach  the  base  without  ever  admitting  of  an  actual 
intersection  .with  it. 

329.  This  curve  was  invented  by  Nicomedes,  a  Greek  geo 
meter,  who  flourished  about  200  years  B.  c.  He  called  it  the 
conchoid,  from  a  Greek  word  signifying  "a  shell":  it  was 
employed  by  him  in  solving  the  problems  of  the  duplication 
of  the  cube,  and  the  trisection  of  an  angle.  To  show  how 
the  curve  may  be  applied  to  the  latter  problem,  let  BAG  be 

the  angle  to  be  trisected  (Fig. 
8):  then  if  CDE  be  drawn  so 
that  the  exterior  segment  DE 


T3 

A  J?  shall  be  equal  to  the  radius  DA; 

it  is  immediately  seen  that  the  arc  DG  is  one-third  of  the  arc 
BC.  Now  it  is  utterly  impossible  so  to  draw  CDE  by  the  aid 
of  the  common  geometry  alone,  that  is,  by  employing  simply 
the  straight  line  and  circle,  but  it  may  easily  be  done  by  re 
sorting  to  the  conchoid.  Let  C  be  the  pole  of  the  inferior 
conchoid,  BE  the  asymptote  or  base,  and  AC  the  modulus, 
then  the  intersection  of  the  curve  with  the  circle  plainly  gives 
the  desired  point  D.  The  superior  conchoid  may  be  employed 
for  the  same  purpose.  The  polar  equation  of  the  conchoid  is 
easily  found,  and  is,  r  =  a  sec  6  -f  b. 

330.  In  the  discussion  of  the  two  preceding  curves,  we  have 
had  occasion  to  allude  to  the  famous  problem  of  the  duplica 
tion  of  the  cube,  the  origin  of  which  is  well  known.  As  it 
deserves  some  notice,  on  account  of  the  celebrity  to  which  it 
attained  among  the  ancient  geometricians,  we  shall  here  intro 
duce  a  very  simple  solution  of  it,  by  means  of  Conic  Sections. 
Let  a  denote  the  edge  of  the  given  cube,  and  x  that  of  the 

t»,...,J    U,      ant.     ft  t 


fltr  $  c,e A:, 


CHAP.  VI.] 


ANALYTICAL  GEOMETRY. 


231 


required  cube ;  then  the  solution  of  the  problem  requires  the 

determination  of  re  so  as  to  satisfy  the  condition,  x3==  2a3 (1). 

Xo',v  as  \vc  may  regard  (1)  as  the  final 

equation  resulting  from  the  elimination 

of  y  between  two  other  equations  y  = 

f  (x),  and   y  =  F  (x\  and  if  we  can 

determine  what   these   equations    are, 

and  then  construct  the  curves  defined 

by  them,  the  abscissa  x  of  their  point  of  intersection  will  be 

the  edge  of  the  required  cube.     To  effect  this,  multiply  (1)  by 

.r,   and   we   get,    z4  =  2a3x (2).     Next,    assume   y-  = 

2ax (3).     Combining  (2)  and  (3)  we  obtain  x*  =  a?y~, 

or,  x-  =  ay (4).     The  required  equations  are  then  (3) 

and  (4) ;  (3)  representing  the  parabola  AYP  (Fig.  9),  and  (4) 
representing  the  parabola  ASP,  the  parameter  of  the  first 
being  double  that  of  the  second.  The  abscissa  AX  of  their 
point  of  meeting  is  the  edge  of  the  required  cube. 

The  Lemniscata  of  Bernouilli.     (Fig.  10.) 
331.  This  curve  was  invented  by  James  Bernouilli.     It  is 
the  locus  of  the  intersections  of  tangents  to  the  equilateral 
hyperbola  with    perpendiculars 
to  them  from  the  centre.     Its 
polar    equation    is,   r2  =  a2  cos 

29 (1).    When  6  =  o,  (1) 

gives   r  =  a,  which   designates 

the  point  A;  as  6  increases  r 

diminishes,  and  when  &  =  45°,  r  =  o,  showing  that  the  curve 

passes  through  the  pole.    If  6  >  45°  but  <  135°,  2d  >  90°  and 

<270°,  so  that  cos  2d  is  negative  and  r  imaginary.     Drawing 

ihen  the  two  lines  SPR  and  S'PK',  making  respectively  angles 


ANALYTICAL  GEOMETRY.  [CHAP.  VI. 

of  45°  and  135°  with  PA,  the  curve  will  not  exist  in  the 
angles  SPS'  and  RPR',  but  will  lie  in  both  the  angles  SPR' 
and  S'PR.  From  0=135°  to  0  =  180°,  r  increases ;  for 
6  =  180°,  r  =  a,  giving  the  point  A'.  From  0  =  180°  to 
&  =  225°,  r  diminishes,  and  for  &  =  225°,  r  =  o.  From 
&  =  225°  to  6  =  315°,  r  is  imaginary.  From  0  =  315°  to 
4  =  360°,  r  increases  till  &  =  360°,  when  r  ==  #,  giving  the 
point  A.  The  shape  of  the  curve  is  that  of  the  figure  8,  as 
shown  in  the  diagram.  By  the  aid  of  the  transcendental 
analysis,  this  curve  is  found  to  be  quadrable,  the  entire  area 
which  it  encloses  being  equivalent  to  the  square  on  the  semi- 
axis  PA. 

Parabolas  of  the  Higher  Orders. 

332.  This  name  designates  a  class  of  curves  represented  by 

the  equation  ym  =  am~nxn (1),  or  by  ym+n  =  amxn (2), 

the  essential  condition  being  that  the  sum 
of  the  exponents  be  the  same  in  each 
member.  When  m  =  2,  and  n  =  1, 
equation  (1)  becomes,  y*  =  ax.  the  com 
mon  or  conical  parabola.  When  m  —  2, 
and  n  =  3,  (1)  gives  us  ?/2  =  a~'#3,  which  represents  the  semi- 
cubical  parabola,  so  named  because  its  equation  may  be  written, 
#|  =  a±y.  The  form  of  this  curve  is  shown  in  Fig.  11.  It 

is  remarkable  as  being  the  first  curve 
which  was  rectified,  that  is,  the  length 
-  of  any  portion  of  it  was  shown  to 
be  equal  to  a  number  of  the  common 
rectilinear  unit.  Its  polar  equation 
is,  r  =  a  tang  2d,  sec  0.  When  m  =  1,  and  n  =  3,  (1)  gives 
a2?/  =  x3,  which  represents  the  cubical  parabola.  Its  form  is 


Fia  12. 


CHAP.  VI.] 


AXALYTICAL  GEOMETRY. 


233 


exhibited  in  Fig.  12.    Its  polar  equation  is  easily  found  to  be, 
r2  =  a\  tang  69  sec  2d. 

333.  Transcendental  Curves.  —  This  appellation  designates 
a  class  of  curves  whose  equations  are  not  purely  algebraic, 
and  are  so  called  because  it  transcends  the  power  of  analysis 
to  express  the  degree  of  the  equation.     As  many  of  these 
curves  are  found  to  possess  remarkable  mechanical  properties, 
•we  shall  proceed  to  the  consideration  of  some  of  the  most 
noted  of  them,  beginning  with 

The  Logarithmic  Curve.     (Fig.  13.) 

334.  This  curve  derives  its  name  from  one  of  its  co-ordinates 
being  the  logarithm  of  the  other.     If  the  axis  of  x  be  taken 
as  the  axis  of  numbers,  that  of 

y  will  be  the  axis  of  logarithms  ; 
and  laying  off  any  numbers,  1, 
2,  3,  4,  etc.,  on  AX,  the  loga 
rithms    of    these    numbers,    as 
found  in  the  Tables   of  Loga 
rithms,   estimated   on   parallels    r 
to  the  axis  of  y,  will  be  the  cor-    A 
responding    ordinates     of     the 
curve. 

From  what  has  been  said,  the 
equation  of  the  curve  is,  y  = 
log  x ;  or,  calling  a  the  base  of 
the  system  of  logarithms,  we 
have,  x  =  ay. 

If  the  base  of  the  system  be  changed,  the  values  of  y  will 
vary  for  the  same  value  of  x;  hence,  every  system  of  loga 
rithms  will  produce  a  different  logarithmic  curve.     The  equa- 
20*  2rc 


234  ANALYTICAL  GEOMETRY.  [CHAP.  VI. 

tion  x  =  avj  enables  us  at  once  to  construct  points  of  the 
curve;  for,  making  successively,  «/ =  0,  y  =  J-,  y  —  f,  etc., 
we  find,  x  =  1,  x  =  \/a,  x  =  \/  a3,  etc.  As  y  =  0,  gives  x  =  1, 
whatever  be  the  system  of  logarithms.,  it  follows  that  every 
logarithmic  curve  cuts  the  axis  of  numbers  at  an  unit's  dis 
tance  from  the  origin. 

335.  If  a  ^>  1,  all  values  of  x  greater  than  unity  will  give 
real  and  positive  values  for  y ;  the  curve,  therefore,  extends 
indefinitely  above  the  axis  of  numbers.     For  values  of  x  less 
than  unity,  y  becomes  negative,  and  increases  as  x  diminishes  ; 
and  when  x  =  o,  y  =  —  oo.     The  curve,  then,  extends  indefi 
nitely  below  the  axis  of  numbers,  and  as  it  approaches  con 
tinually  the  axis  of  logarithms,  this  axis  is  an  asymptote  to 
the  curve.     If  x  be  negative,  y  becomes  imaginary ;  the  curve 
is,  therefore,  limited  by  the  axis  of  logarithms. 

336.  If  a  <  1,  the  situation  of  the  curve  is  reversed,  and  is 
such  as  is  represented  by  the  dotted  line  in  the  figure. 

337.  Taking  the  axis  of  y  for  the  axis  of  numbers,  that  of 
x  would  be  the  axis  of  logarithms,  and  the  curve  would  enjoy, 
relatively  to  this  system,  the  same  properties  which  have  been 
demonstrated  above. 

338.  This  curve  was  invented  by  James  Gregory.    Huygbens 
discovered  that  if  PT  be  a  tangent  meeting  AY  at  T,  YT  is 
constant  and  equal  to  the  modulus  of  the  system.     Also  that 
the  whole  area  PYV&P  extending  indefinitely  towards  V,  is 
finite,  and  equal  to  twice  the  triangle  PYT ;  and  that  the  solid 
described  by  the  revolution  of  the  same  area  about  AY,  is  1J 
times  the   cone   generated   by  revolving   the    triangle   PYT 
about  AY. 


CHAP.  VI.] 


ANALYTICAL  GEOMETRY. 


235 


The  Cycloid.     (Fig.  14.) 

339,  If  a  circle  QMG  be  rolled  along  the  line  AB,  any 
point  M  of  its  circumference  will  describe  a  curve  AMKL, 
•which  is  called  a  Cycloid.  This  is  the  curve  which  a  nail  in 
the  rim  of  a  carriage-wheel  describes  in  the  air  during  the 
motion  of  the  carriage  on  a  level  road.  The  curve  derives  its 
name  from  two  Greek  words  signifying  "circle-formed."  The 
line  AL  over  which  the  generating  circle  passes  in  a  single 
revolution  is  called  the  base  of  the  cycloid,  and  if  I  be  the 
middle  point  of  AL,  the  point  K  is  called  the  vertex,  and  the 
line  KI  the  altitude  or  axis  of  the  curve.  To  find  its  equation, 


Fig.  14. 


CL, 


let  K  be  the  origin  of  co-ordinates ;  put  Kn  =  x,  wM  =  y, 
and  SI,  the  radius  of  the  generating  circle,  =  a.  Then  we  have, 

Mft  =  M?tt  -f  mn (1). 

And 

Mm  =  QI  =  AI  — AQ (2). 

Now  from  the  mode  of  generation,  we  have,  AQ  =  arc  MQ  =. 
arc  7?? I;  and  AI  =  semi-circumference  IniK.  Hence  (2) 
becomes,  M??z  =  ImK  —  arc  ml=  arc  Km,  and,  consequently, 
(1)  becomes, 


y  =  arc  Km  +  mn  =  arc  Km  -f  ^/Kn  X  nl  —  arc  Km  + 
</2ax—x* (3). 


236  ANALYTICAL  GEOMETRY.  [CHAP.  VI. 

Now  we  have  arc  Km  =  a  circular  arc  whose  radius  is  a  and 

ver  sin  x  =  a  (an  arc  whose  radius  is  unity  and  ver  sin  -)  ;  or, 

~lx 
introducing  the  notation,  ver  sin     -  to  signify  "  the  arc  whose 

x 
versed  sine  is  — ,"  (3)  may  be  written, 


y  =  a  ver  sin     -  +  >/  2ax  —  a2 (4) 

for  the  equation  of  the  cycloid. 

The  equation  of  the  curve  is  frequently  to  be  met  with 
referred  to  A  as  an  origin,  with  AB  as  the  axis  of  x,  and  AY 
the  axis  of  y.  Its  equation  then  is, 

~~'  y  

x  =  a  ver  sin     —  —  </2ay  —  y* (5). 

The  cycloid  is  not,  of  course,  terminated  at  the  point  L,  but 
as  the  generating  circle  moves  on,  similar  cycloids  are  described 
along  AB  produced.  The  points  A  and  L,  when  the  consecu 
tive  curves  of  the  series  join  each  other,  are  termed  cusps  or 
points  of  cusp  —  the  designation  not  being  restricted  to  the 
cycloid  alone,  but  used  as  one  applied  generally  to  a  similar 
union  between  the  branches  of  any  curve.  We  have  already 
had  examples  of  such  points  in  the  cissoid  and  semi-cubical 
parabola. 

340.  The  cycloid,  if  not  first  imagined  by  Galileo,  was  first 
examined  by  him ;  and  it  is  remarkable  for  having  engaged 
the  attention  of  the  most  eminent  mathematicians  of  the  seven 
teenth  century. 

341.  With  the  exception  of  the  Conic  Sections,  no  known 
curve  possesses  so  many  beautiful  and  useful  properties  as  the 
cycloid.     Some   of  these   are,  that   the   area  AMKwIA,  is 
equivalent  to  that  of  the  generating  circle;  that  the  entire 


CHAP.  VI.] 


ANALYTICAL  GEOMETRY. 


237 


area  AKLA,  is  equivalent  to  three  times  that  of  the  generating 
circle  ;  that  the  tangent  MG  is  parallel  to  the  chord  mK ;  that 
the  length  of  the  arc  MK  is  double  that  of  the  chord  Kiw, 
and  consequently  the  entire  perimeter  AMKCL  is  four  times 
the  diameter  of  the  generating  circle ;  that  if  the  curve  be 
inverted,  and  two  bodies  start  along  the  curve  from  any  two 
of  its  points,  as  A  and  M,  at  the  same  time,  they  will  reach 
the  vertex  K  at  the  same  moment ;  and  if  a  body  falls  from 
one  point  to  another  point  not  in  the  same  vertical  line,  its 
path  of  quickest  descent  is  not  the  straight  line  joining  the 
two  points,  but  the  arc  of  an  inverted  cycloid  connecting  them. 


On  account  of  these  last  two  properties,  the  cycloid  is  called  the 
tautochronal  and  bracfiystochronal  curve,  or  curve  of  equal 
and  swiftest  descent. 

342.  Instead  of  the  generating  point  being  on  the  circum 
ference  of  the  circle,  it  may  be  anywhere  in  the  plane  of  that 


238  ANALYTICAL  GEOMETRY.  [CHAP.  TI. 

circle,  either  within  or  without  the  circumference.  In  the 
former  case,  the  curve  is  called  the  Prolate  Cycloid,  or  Trochoid 
(Fig.  15) ;  in  the  latter  case,  the  Curtate,  or  shortened, 
Cycloid  (Fig.  16). 

343.  To  find  the  equations  of  these  curves,  let  K  (Figs.  15 
and  16)  be  the  origin  of  co-ordinates.  Put  KM  =  x,  MP  =  y, 
KO  =  a,  AO  =  ma,  <  AOR  =  <p. 

Then  from  the  figure,  MP  =  FC  +  QM  =  arc  AR  -f  QM. 


.-.y  —  may+a  sin  9,  or,  y=maver  sin      —  -f  ^/2ax  —  x\ 

which  equation  will  represent  the  common  cycloid  if  m  =  1  ; 
the  prolate  cycloid  when  m  >  1 ;  and  the  curtate  cycloid  when 
ro<l. 

344.  The  class  of  cycloids  may  be  much  extended  by  sup 
posing  the  base  on  which  the  generating  circle  rolls,  to  be  no 
longer  a  straight  line,  but  itself  a  curve  :  thus,  let  the  base  be 
a  circle,  and  let  another  circle  roll  on  the  circumference  of  the 
former ;  then  a  point  either  within  or  without  the  circumference 
of  the  rolling  circle  will  describe  a  curve  called  the  EpitrocJioid ; 
but  if  the  describing  point  is  on  the  circumference,  it  is  called 
the  Epicycloid. 

345.  If  the  revolving  circle  roll  on  the  inner  or  concave 
side  of  the  base,  the  curve  described  by  a  point  within  or  with 
out  the  revolving  circle  is  called  the  Hypotroclwid  ;  and  when 
the  generating  point  is  on  the  circumference  of  the  rolling 
circle,  the  curve  is  called  the  Hypocycloid. 

346.  To  obtain  the  equations  of  these  curves,  we  shall  find 
that  of  the  EpitrocJioid,  and  then  deduce  the  rest  from  it. 
(Fig.  17.) 

Let  C  be  the  centre  of  the  base  ED0.  and  B  the  centre  of 
the  revolving  circle  DF  in  one  of  its  positions :  CAM  the 


CHAP.  YL]  ANALYTICAL  GEOMETRY. 

straight  line  passing  through  the  centres  of  both  circles  at  the 
commencement  of  the  motion ;  that  is,  when  the  generating 


A  °N   M 


point  P  is  nearest  to  C,  or  at  A.  Let  CA  be  the  axis 
of  x;  CM  -  x,  MP  =  y,  CD  =  a,  DB  =  b,  BP  =  ml,  and 
>  ACB  =  <p. 

Draw  BN  parallel  to  MP,  and  PQ  parallel  to  EM.     Then, 
since  every  point  in  DF  has  coincided  with  the  base  AD,  we 

ctq> 
haye  DF  =  a<p,  and  angle  DBF  =  -r  ;  also  angle 


Now  CM  =  CN  4-  NM  =  CB  cos  BCN  +  PB  sin  PBQ 

(a  4-  b)  cos  9  4  mb  sin  ( — ? —  9  — 


And, 


-a  4-  b 


MP  =  BN —  BQ  =  (a  4  b)  sin  9  —  nib  cos  ( — ^—  9  —  ^ 
or, 

I        ,     JA  I  a  +   b 

x  =  (a  4  6)  cos  9  —  mo  cos  — r —  9, 

and,  ?/  =  (a  4  3)  sin  9  —  mb  sin  — r —  9 
Such  are  the  equations  which  represent  the  Epitrochoid.. 


(1) 


240 


ANALYTICAL  GEOMETRY.  [CHAP.  VI 


Those  for  the  Epicycloid  are  found  by  putting  b  for  mb 

in  (1). 

a  -f  b 
.-.  x  —  (a  +  b)  cos  9  —  b  cos  —  7  — 


an(J  y  =  (a  +  0)  sm  <p  —  6  sin  —  » 

Those  for  the  Hypotrochoid  may  be  obtained  by  writing  —  b 
for  b  in  (1),  and  those  for  the  Hypocycloid  are  found  by  putting 
—  b  for  both  b  and  ra&  in  (1). 

347.  The  '  elimination  of  the  trigonometrical  quantities  is 
possible,  and  gives  finite  algebraic  equations  whenever  a  and  b 
are  in  the  ratio  of  two  integral  numbers.  For  then  cos  9, 

cos  —  T~  9,  sin  9,  etc.,  can  be  expressed  by  trigonometrical  formu 
las  in  terms  of  cos  ^  and  sin  4,,  when  -^  is  a  common  submultiple 
of  9  and  —  7  —  9  ;  and  then  cos  ^  and  sin  4*  may  be  expressed 

in  terms  of  x  and  y.  Also  since  the  resulting  equation  in  x 
and  y  is  finite,  the  curve  does  not  make  an  infinite  series  of 
convolutions,  but  the  revolving  circle,  after  a  certain  number 
of  revolutions,  is  found  having  the  generating  point  exactly 
in  the  same  position  as  at  first,  and  thence  describing  the  same 
curve  line  over  again. 

For   example,  let  a  =  6,  the   equations  to  the  Epicycloid 
become, 

x  =  a  (2  cos  <p  —  cos  2  9),  y  =  a  (2  ,sin  9  —  sin  2  9)  ; 


or, 


x  =  a  (2  cos  9  —  2  cos29  +  1) 


} (3) 


y  =  2a  sin  9  (1  —  cos  9) 
From  the  first  of  equations  (3)  we  find  the  value  of  cos  tp ; 
and  from  the  second,  that  of  sin  9,  and  then  adding  together 
the  values  of  cos29  and  sin29,  and  reducing  we  vrf- 


CHAP.  VI.]  ANALYTICAL  GEOMETRY.  241 

2: 


or, 

(3?  -f-  y*  —  a2)2  —  4a2  \  (x  —  of  -f-  y*  }  =  o. 

This  curve,  from  its  heart-like  shape,  is  called  the  Cardioide. 
If  the  origin  be  transferred  to  A,  the  polar  equation  of  this 

curve  becomes, 

r  =  2a  (1  —  cos  d). 

348.  If  b  =  Q,  the  equations  of  the  hypocycloid  become, 

x  =  a  cos  <p,  and  y  —  o  ;  i.  e.,  the  curve  reduces  to  the  diameter 
of  the  circle  ACE.  Under  the  same  supposition,  the  hypotro- 
choid  reduces  to  an  Ellipse  whose  axes  are  a  (m  +  1)  and 

Spirals. 

349.  S}nrals   comprise   a   class   of  transcendental   curves 
which  are  remarkable  for  their  form  and  properties.     They 
were  invented  by  the  ancient  geometricians,  and  were  much 
used  in  architectural  ornaments.     The  principal  varieties  are, 
the  Spiral   of  Archimedes,  the    Hyperbolic,  Parabolic,  and 
Logarithmic  Spirals,  and  the  Lituus. 

Spiral  of  Archimedes.     (Fig.  18.) 

350.  If  a  line  A0  revolve  uniformly  around  a  centre  A, 
at  the  same  time  that  one  of  its  points  commencing  at  A,  with 
a  regular  angular  and  outward  motion,  describes  a  curve  AM0, 
and  is  found  at  o,  when  A0  has  completed  one  entire  revolu 
tion,  and  at  X  at  the  end  of  the  second  revolution,  and  so  on, 
the  curve  AMoM'X,  will  be  the  /Spiral  of  Archimedes. 

From  the  nature  of  this  generation,  it  follows  that  the  ratio 
of  the  distance  of  each  of  its  points  from  the  point  A,  to  the 
21  2F 


242 


ANALYTICAL  GEOMETRY. 


[CHAP.  VI 


length  of  the  line  Ao,  will  be  equal  to  that  of  the  arc  passed 
over  by  the  point  o  from  the  commencement  of  the  revolution, 


Fig.  18. 


to  the  entire  circumference  ;  or,  for  any  point  M',  we  shall 

have, 

AM'      o&o  + 


AN  oGo        > 

and  making  0GN  =  6,  AM  =  r,  AN  =  1,  the  circumference 
cGo  will  be  denoted  by  Zie,  and  the  equation  of  the  spiral  be- 

A 

comes,  r  =  9-  .     The  variables  in  this  equation  are  those  of 

polar  co-ordinates.  The  point  A  is  the  pole  or  eye  of  the 
spiral,  AM  the  radius  vector,  and  the  angle  subtending  oGN 
the  variable  angle. 

351.  The  curve  which  has  just  been  considered  is  a  par 
ticular  case  of  the  class  of  spirals  represented  by  the  general 
equation,  r  —  abn,  where  a  and  n  represent  any  quantities 
whatsoever. 

The  Hyperbolic  Spiral.     (Fig.  19.) 

352.  If  in  the  general  equation,  r  =  a6n,  we  have  n  =  —  1, 

the  resulting  equation  r  =  -  ,  will  be  that  of  the  Hyperbolic 


CHAP.  VI.] 


AXALYTICAL  GEOMETRY. 


Spiral,  called  also,  the  reciprocal  spiral.     This  curve  has  an 
asymptote. 

In  fact,  if  we  make  successively,  6  =  1,  =  J,  =  J,  etc.,  we 
shall  have  r  =  a,  =  2a,  =  3a,  etc.,  which  shows  that  as  the 


Pig.  19. 


spiral  departs  from  the  point  A,  it  approaches  continually  the 
line  DE  drawn  parallel  to  AO,  and  at  a  distance  AB  =  a.  For, 
drawing  PM  perpendicular  to  AB,  we  have, 


PM  =  r  sin  MAP  =  r  sin  &  =  a 


sin  6 


when  r  is  replaced  by  its  value  -^.     This  value    of  PM  ap 
proaches  more  and  more  to  a  as  6  diminishes,  and  when  6  ig 

sin  & 
very  small,  —^-=1,  and   PM=a;   DE    is   therefore    an 

asymptote  to  the  curve.  If  6  be  reckoned  from  AB',  we 
shall  have  a  similar  spiral  to  which  DE'  will  be  an  asymptote. 
This  curve  takes  its  name  from  the  similarity  of  its  equation 
to  that  of  the  hyperbola  referred  to  its  asymptotes ;  r$  =  a, 
being  that  of  the  spiral,  and  xy  =  M2,  that  of  the  hyperbola. 

The  Parabolic  Spiral     (Fig.  20.) 

353.  This  spiral  is  generated  by  wrapping  the  axis  AX  of 
a  parabola  around  the  circumference  of  a  circle.    The  ordinates 


244 


ANALYTICAL  GEOMETRY. 


[CHAP.  VI. 


PM,  P'M',  will  then  coincide  with  the  prolongations  of  the 
radii  ON,  ON' ;  and  the  abscissas  AP,  AP',  of  the  parabola, 

will  coincide  with  the  arcs  AN,  AN', 
etc.     AQQ'Q",  etc.,  is  the  spiral. 

The  equation  of  the  parabola  being 
2/2  =  2px  ;  we  have,  QN  =  r  —  b  =  y, 
b  being  the  radius  of  the  circle ;  and 
AN  =  6  =  x.  The  equation  of"  the 
spiral  then  becomes,  (r  —  Vf  — 
2pQ  =  a&j  by  making  2p  =  a.  If  the 
origin  of  the  curve  be  at  the 
centre  of  the  circle,  b  —  o,  its 
a&. 


rig.  20. 


equation  becomes,  r* 


The  Logarithmic  Spiral.     (Fig.  21.) 

354.  The  equation  of  this  curve  is,  6  =  log  r,  or  r  =  ae, 
when  a  is  the  base  of  the  system  of  logarithms  used.     Making 

0  =  0,  we  get,  r  —  1.  The  curve 
therefore  passes  through  the  point 
0.  As  r  increases,  d  increases 
also  ;  there  is  therefore  an  infinite 
number  of  revolutions  about  the 
circle  OGN.  When  r  <  1,4  be 
comes  negative,  and  its  values 
give  the  part  of  the  curve  within 
the  circle  OGN.  As  r  diminishes,  6  increases,  and  when 
r  =  o,  6  =  —  ex.  The  spiral  therefore  continually  approaches 
the  pole,  but  never  reaches  it. 


Fig.  21. 


CHAP.  VI.] 


AXALYTICAL  GEOMETRY. 


245 


The  Lituus.     (Fig.  22.) 

355.  The  Lituus,  or  trumpet,  is  a  spiral  represented  by  the 
equation,  r~&  =  a2.     Its  form 

is  exhibited  in  the  diagram. 

. 

The  fixed  axis  is  an  asymp 
tote,  and  the  curve  makes  an  infinite  series  of  convolutions 
around  the  pole  without  attaining  it. 

Remark. 

356.  In  the  discussion  of  curves  there  is  one  point  deserving 
consideration,  namely :  it  will  often  happen  that  the  algebraical 
equation  of  a  curve  is  much  more  complicated  than  its  polar 
equation ;  the  conchoid  is  an  example.     In  these  cases  it  is 
advisable  to  transform  the  equation  from  algebraic  to  polar 
co-ordinates,  and  then  trace  the  curve  by  means  of  the  polar 
equation. 

We  subjoin  several  examples  as  an  exercise  for  the  student. 

1.  (z2  -f  y^  =  Zaxy  ;  which  gives,  r  =  a  sin  26. 

2.  (z2  +  y-J  =  Za'xy. 

3.  x*  +  y*  =  a(x  —  y). 

4.  (V  +  2/2)2  =  a2(z2  —  ?/2). 

357.  In  many  indeterminate  problems  we  shall  find  that 
polar  co-ordinates  may  be  very  usefully 
employed.  For  example  :  Let  the  corner 
of  the  page  of  a  book  be  turned  over  into 
the  position  BCP  (Fig.  23),  and  in  such 
manner  that  the  area  of  the  triangle  BCP 
be  constant ;  to  find  the  locus  of  P.  Let 
AP  =  r,  >  PAG  =  6,  and  area  ABC  =  a2 

Then 
21* 


246  ANALYTICAL  GEOMETRY.  [CHAP.  VI. 

f  T1        CC* 

AE  ==  g,  AE  =  AC  cos  6,  AE  =  AB  sin  6,  .-.  ^  ==  -^  sin  6  cos  0 ; 

or, 

r2  =  a2  sin  23, 

for  the  required  equation. 

358.  In  some  cases  it  may  be  advisable  to  exchange  polar 
co-ordinates  for  algebraic  ones,  the  formulas   for  which  are 
(when  the  new  system  is  rectangular), 

y  x  

sin  d  =  — ,  cos  6  =  —,  and  r  =  \/x*  +  y2. 

f  f  V 

359.  We  have  now  given  a  sufficiently  extensive  discussion 
of  the  curves  of  the  higher  orders,  and  shall  next  proceed  to 
give  a  few  examples  to  be  investigated  by  the  student  himself, 
in  order  that  he  may  become  entirely  familiar  with  the  appli 
cation  of  the  principles  already  laid  down.    And  here  we  may 
observe,  that  while  the    methods   here   given  will  ordinarily 
prove  sufficient  for  determining  the  general  outline  and  form 
of  most  curves,  yet  there  are  many  which  yield  a  complete 
solution  only  when  subjected  to  the  exhausting  processes  of 
the  higher  calculus ;  and  indeed  its  aid  is  almost  indispensable 
for  arriving  at,  and  thoroughly  discussing,  many  of  the  most 
valuable  and  beautiful  properties  of  some  of  the  curves  we 
have  already  considered.     The  methods  of  Analytical  Geome- 
*ry  are  not,  however,  on  this  account,  less  deserving  the  study 
and  time    of  the  pupil,  since  the  expedients    of  the   higher 
analysis    are   based    upon    them ;    presupposing,  and   indeed 
requiring,  a  familiar  acquaintance  with  their  details. 

EXAMPLES. 

2.  (a  —  xfif  =  x  (b  —  x)2. 


CHAP.  VI.]  ANALYTICAL  GEOMETRY.  247 

3.  axy  =  x5  —  a3. 

4.  (x-  —  I)y*=  Zx  —  x\ 

5.  y*  —  2x-yz  —  z4  -f  1  =  o, 

6.  x'Y~  —  xy-  =  1. 

7.  xy-  +  yx-  =  1. 

8.  f  —  x-y-  =  x\ 

9.  y-  =  x3  —  x4. 

10.  (1  -f  x)y*=  1. 

11.  (l-x-)y=l. 

12.  y  =  x  ±  a;  v/2;. 

13.  y  =  ^2  ±  -,. 


15.  r  =  cos  6  -f  2  sin 

2 

16-  r  =  TTtSjfi' 

17.  ^  =  -r— 0- 

sin  25  d 

18.  r  =  tang  5. 

19.  r  =  1  4-  2  cos  0. 

20.  r  =  — ,-r. 


21.  r  = 


22.  r 


cos 
1  +  sin  d 
1  —  sinl* 
1 


3  tang  dJ 
23.  r2  =  a-  sec2  4  (1  —  sin* 


24S  ANALYTICAL  GEOMETRY.  [CHAP.  VII 


CHAPTER  VII. 

OF  SURFACES  OF  THE  SECOND  ORDER. 

360.  SURFACES,  like  lines,  are  divided  into  orders,  according 
to  the  degree  of  their  equations.     The  plane,  whose  equation 
is  of  the  first  degree,  is  a  surface  of  the  first  order. 

361.  We  will  here  consider  surfaces  of  the  second  order, 
the  most  general  form  of  their  equation  being 

Az2  +  Ay  +  A"*2  +  Ryz  +  E'xz  +  Wxy  -f  Cz  + 
C'y  +  C"x  +  F  =  o.         (I) 

Since  two  of  the  variables,  x,  y,  z,  may  be  assumed  at 
pleasure,  if  we  find  the  value  of  one  of  them,  as  z,  in  terms 
of  the  other  two,  we  could,  by  giving  different  values  to  x 
and  y,  deduce  the  corresponding  values  of  z,  and  thus  deter 
mine  the  position  of  the  different  points  of  the  surface.  But 
as  this  method  of  discussion  does  not  present  a  good  idea  of 
the  form  of  the  surfaces,  we  shall  make  use  of  another  method, 
which  consists  in  intersecting  the  surface  by  a  series  of 
planes,  having  given  positions  with  respect  to  the  co-ordinate 
axes.  Combining  then  the  equations  of  these  planes  with 
that  of  the  surface,  we  determine  the  curves  of  intersections 
whose  position  and  form  will  make  known  the  character  of 
the  given  surface. 

362.  To  exemplify  th.a  method,  take  the  equation 

x2  +  v2  +  ;;2  =  R2, 


CHAP.  VII.]  ANALYTICAL  GEOMETRY.  249 

and  let  this  surface  be  intersected  by  a  plane,  parallel  to  the 
plane  of  xy ;  its  equation  will  be  of  the  form  (Art.  73), 

z  =  a, 

and  substituting  this  value  of  z,  in  the  proposed  equation, 

we  have 

x2  -f  y2  -  R2  —  a\ 

for  the  equation  of  the  projection  of  the  intersection  of  the 
plane  and  surface  on  the  plane  of  xy.  It  represents  a  circle 
(Art.  133),  whose  centre  is  at  the  origin,  and  whose  radius 
is  \/R2  —  a2.  This  radius  will  be  real,  zero,  or  imaginary, 
according  as  a  is  less  than,  equal  to,  or  greater  than  R.  In 
the  first  case  the  intersection  will  be  the  circumference  of  a 
circle,  in  the  second  the  circle  is  reduced  to  a  point,  and  in 
the  third  the  plane  does  not  meet  the  surface. 

363.  The  proposed  equation  being  symmetrical  with  respect 
to  the  variables  x,  y,  z,  we  shall  obtain  similar  results  by 
intersecting  the  surface  by  planes  parallel  to  the  other  co 
ordinate  planes.  It  is  evident,  then,  that  the  surface  is  that 
of  a  sphere. 

332.  The  co-ordinate  planes  intersect  this  surface  in  three 
equal  circles,  whose  equations  are, 


364.  We  may  readily  see  that  the  expression  Vx2  -f  t/2  +  za 
represents  a  spherical  surface,  since  it  is  the  distance  of  any 
point  in  space  from  the  origin  of  co-ordinates  (Art.  75),  and 
as  this  distance  is  constant,  the  points  to  which  it  corresponds 
are  evidently  on  the  surface  of  a  sphere,  having  its  centre  at 
the  origin  of  co-ordinates. 


250  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

365.  The  discussion  has  been  rendered  much  more  simple, 
by  taking  the  cutting  planes,  parallel  to  the  co-ordinate 
planes,  since  the  projections  of  the  intersections  do  not  differ 
from  the  intersections  themselves.  Had  these  planes  been 
subjected  to  the  single  condition  of  passing  through  the  origin 
of  co-ordinates,  the  form  of  their  equations  would  have  been 

Ax  +  By  +  Cz  =  o  ; 

and  combining  this  with  the  proposed  equation,  we  should    V%  ^' 
have, 

(A2  +  C2)  x2  +  2ABxy  +  (B2  +  C2)  y1  =  R2C2, 

which  is  the  equation  of  the  projection  of  the  intersection  on 
the  plane  of  xy.  This  projection  is  an  ellipse,  but  we  can 
readily  ascertain  that  the  intersection  itself  is  the  circum 
ference  of  a  circle,  by  referring  it  to  co-ordinates  taken  in 
the  cutting  plane. 

366.  We  may  in  the  same  manner  determine  the  character 
of  any  surface,  by  intersecting  it  by  a  series  of  planes,  and 
it  is  evident  that  these  intersections  will,  in  general,  be  of 
the  same  order  as  the  surface,  since  their  equations  will  be 
of  the  second  degree. 

367.  Before  proceeding  to  the  discussion  of  the  general 
equation 

Az*  -f  Ay  +  A"  a;2  +  Eyz  +  E'xz  +  E"xy  +  Cz  + 
C'y  +  C"x  +  F  =  o, 

let  us  simplify  its  form,  by  changing  the  origin,  so  that  we 
have,  between  the  two  systems  of  co-ordinates,  the  relations 
(Art.  114), 

y~ !/  +  h  *  =  *  +  /' 


CHAP.  VII.]  ANALYTICAL  GEOMETRY.  251 

As  a,  3,  7,  are  indeterminate,  we  may  give  such  values  to 
them  as  to  cause  the  terms  of  the  transformed  equation 
affected  with  the  first  power  of  th'e  variables  to  disappear. 
This  requires  that  we  have 

2A7    +  B/3  +  B'a  +  C    -  o, 

2A'/3  +  B"a  +  B7    -f  C'  =  o, 

2A"a  +  B'7  +  B"/3  +  C"=  o;         (2) 

and,  representing  all  the  known  terms  in  the  transformed 
equation  by  L,  it  becomes 

Az'2  +  Ay2  +  AV2  +  B~y  +  E'z'x'  +  B"x'y'  +  L  =  o.     (3) 

As  all  the  terms  in  this  equation  are  of  an  even  degree, 
its  form  will  not  be  changed,  if  we  substitute  — x',  — y't 
—  z',  for  -f  x',  -f  y'9  +2'.  If>  then,  a  line  be  drawn  through 
the  origin  of  co-ordinates,  the  points  in  which  it  meets  the 
surface  will  have  equal  co-ordinates  with  contrary  signs. 
This  line  is  therefore  bisected  at  the  origin,  which  will  be 
the  centre  of  the  surface,  if  we  attribute  the  same  significa 
tion  to  this  point  in  reference  to  surfaces  that  wre  have  for 
curves. 

368.  The  equations  (2)  which  determine  the  position  of 
the  centre  being  linear,  they  will  always  give  real  values  for 
a,  (3,  7;  but  the  coefficients  A,  B,  C,  &c.,  may  have  such 
relations  as  to  make  these  values  infinite.  In  this  case  the 
centre  of  the  surface  will  be  at  an  infinite  distance  from  the 
origin,  which  will  take  place  when 

AB' 2  +  A'B2  -f  A"B2  —  BB'B"  —  4AA' A"  =  o,    (D.) 

which  is  the  denominator  of  the  values  of  a,  (3,  7,  drawn 
from  equation  (2)  placed  equal  to  zero. 


252  ANALYTICAL  GEOMETRY.  [CHAP.  VII, 

369.  If  this  condition   be   satisfied,  and  we  have  at  the 
same  time 

C  =  o,     C'  =  o,     C"  =  o, 

the  values  of  a,  /3,  7,  will  no  longer  be  infinite,  but  will  be 
come  — »  which  shows  that  there  will  be  an  infinite  number 
o 

of  centres.  In  this  case  the  surface  is  a  right  cylinder,  with 
an  elliptic  or  hyperbolic  base,  whose  axis  is  the  locus  of  all 
the  centres. 

370.  If  condition  (D)  be  not  satisfied,  but  we  have  simply 

C  =  Of     C'  =  o,     C"  =  0, 

the  values  of  a,  [3,  y,  become  zero,  and  the  centre  of  the  sur 
face  coincides  with  the  origin.  This  is  evident  from  the  fact 
that  equations  (2)  represent  three  planes,  whose  intersection 
determines  the  centre ;  and  these  planes  pass  through  the 
origin  when  C,  C',  C",  are  zero. 

371.  We  may  still  further  simplify   the  equation  (2)  by 
referring  the  surface  to  another  system  of  rectangular  co 
ordinates,  the  origin  remaining  the  same,  so  that  its  equation 
shall  not  contain  the  product  of  the  variables.    The  formulas 
for  transformation  are 

x  =  x"  cos  X  +  y"  cos  X'  +  %"  cos  X", 
y  =  x"  cos  Y  -f  y"  cos  Y'  +  z"  cos  Y", 
*'  =  x"  cos  Z  +  y"  cos  Z'  +  z"  cos  Z", 

with  which  we  must  add  (Arts.  116  and   117), 

cos2X  +cos2Y  +cos2Z  =o, 
cos2X'  -f  cos2Y'  +cos2Z'  =o, 
cos'X"  +  cos2Y"  +  cos2Z"  =  o,  (A) 


Cii^p.  VII.1  ANALYTICAL  GEOMETRY.  253 

cos  X  cos  X'  -f  cos  Y  cos  Y'  4-  cos  Z  cos  Z'  =  o, 
cos  X  cos  X"  4-  cos  Y  cos  Y"  4-  cos  Z  cos  Z"  =  o, 
cos  X'cos  X'  -f  cos  Y'cos  Y"  4-  cos  Z'  cos  Z"  =  o.  (B) 

Equations  (B)  are  necessary  to  make  the  new  axes  rectan 
gular.  These  substitutions  give  for  the  surfa.ce  an  equation 
of  the  form 

Mz"2  +  My2  4-  M'V2  4-  Nz'y  -f  N'z"s"  4-  N'V'y"  +  P  =  o. 

In  order  that  the  terms  in  z"y",  z"x",  x"y",  disappear,  we 
must  have 

N  =  o,    N'  =  o,    N"  =  o. 

Without  going  through  the  entire  operation,  we  can 
readily  form  the  values  of  N,  N',  N",  and  putting  them 
equal  to  zero,  we  have  the  following  equations : 


2A   cos  Z  cos  Z'  4-  B    (cos  Z  cos  Y'  4-  cos  Y  cos  Z') 
4- 2  A'  cos  Y  cos  Y'  4-  B'  (cos  Z  cos  X'  4-  cos  X  cos  Z') 


.    3-. 

4-  2 A"  cos  X  cos  X'  4-  B  '  (cos  Y  cos  X'  4-  cos  X  cos  Y')  ) 


2A  cos  Z  cos  Z"  4-  B  (cos  Z  cos  Y"4-  cos  Y  cos  Z") 
4-2A'  cos  Y  cos  Y"+  B'  (cos  Z  cos  X"4-  cos  X  cos  Z")  )>  =  o.(C) 
4-2A"cos  X  cos  X"4-  B"(cos  Y  cos  X"4-  cos  X  cos  Y") 

2A  cos  Z'  cos  Z"  4-  B  (cos  Z'  cos  Y"4-  cos  Y'  cos  Z") 
4-2 A'  cos  Y'  cos  Y"  4-  B'  (cos  Z'  cos  X' '  4-  cos  X'  cos  Z" )  ^  =  o. 
4-2A"cos  X'  cos  X"  4-  B  '(cos  Y'  cos  X'  4-  cos  X'  cos  Y") 

The  nine  equations  (A)  (B)  (C)  are  sufficient  to  determine 
the  nine  angles  which  the  new  axes  must  make  with  the  old, 
in  order  that  the  transformed  equation  may  be  independent 
of  the  terms  which  contain  the  product  of  the  variables 

22 


254  ANALYTICAL  GEOMETRY.  [CHAP.  VII, 

Introducing  these  conditions,  the  equation  of  the  surface 
becomes 

Mz"2  +  My2  +  MV'2  +  L  =  o,  (4) 

which  is  the  simplest  form  for  the  equations  of  Surfaces  of 
fhe  Second  Order  which  have  a  centre. 

372.  We  may  express  under  a  very  simple  formula,  sur 
faces  with,  and  those  without  a  centre.  For,  if  in  the  general 
equation,  we  change  the  direction  of  the  axes  without  moving 
the  origin,  the  axes  also  remaining  rectangular,  we  may  dis 
pose  of  the  indeterminates  in  such  a  manner  as  to  cause  the 
product  of  the  variables  to  disappear.  By  this  operation  the 
proposed  equation  will  take  the  form 

Mz'2  +  My2  +  MV  +  Kz'  +  K'y  +  KV  +  F  -  o. 

If  now  we  change  the  origin  of  co-ordinates  without 
altering  the  direction  of  the  axes,  which  may  be  done  by 
making 

z'  =  z"  +  a,     y  =  y"  +  a,     z  =  z"  +  a", 

we  may  dispose  of  the  quantities  a,  a ',  a",  in  such  a  manner 
as  to  cause  all  the  known  terms  in  the  transformed  equation 
to  disappear.  This  condition  will  be  fulfilled  if  the  new 
origin  be  taken  on  the  surface,  and  we  have 

Ma2  +  MV/2  +  MV'2  +  Ka  +  KV  +  KV  +  F  =  o.     (5) 
Suppressing  the  accents,  and  making,  for  more  simplicity, 
2Mfl  +  K  =  H,    2MV/  +  K'  =  H',    2M'V  +  K"  =  H", 

every  surface  of  the  second  order  will  be  comprehended  in 
the  equation 

Mz2  +  My  -f  MV  +  Hz  +  ITy  +  H"#  =  o.    (6) 


CHAP.  VII.]  ANALYTICAL  GEOMETRY.  255 

373.  In  order  that  equation  (6)  may  represent  surfaces 
which  have  a  centre,  it  is  necessary  that  the  values  of  a,  a. 
a",  reduce  this  equation  to  the  form  of  equation  (4),  which 
requires  that  the  terms  containing  the  first  power  of  the 
variables  disappear.  This  condition  will  always  be  satisfied, 
if  the  equations 

2Ma  -f  K  =  o,     2M'a  +  K'  =  o,     2M"a"  +  K"  =  o 


give  finite  values  for  «,  a',  a".     These  values  are 
K  Kf  K' 


~2Mf   a  '~     ~' 


and  will  always  be  finite,  so  long  as  M,  M',  M",  are  not  zero 
But  if  one  of  them,  as  M,  be  zero,  the  value  of  a  becomes 
infinite,  and  the  surface  has  no  centre,  or  this  centre  is  at  an 
infinite  distance  from  the  origin. 


Of  Surfaces  which  have  a  Centre. 

374.  We  have  seen  (Art.  340),  that  all  surfaces  of  the 
second  order  which  have  a  centre  are  comprehended  in  the 
equation 

Mz"2  +  M'i"2  +  M".r"2  +L  =  o. 


Suppressing  the  accents  of  the  variables,  we  have 

Mz2  +  My  +  MV  +  L  =  o. 

Let  us  now  discuss  this  equation,  and  examine  more  par 
ticularly  the  different  kinds  of  surfaces  which  it  represents. 

Resolving  this  equation  with  respect  to  either  of  the  vari- 
obles,  we  shall  obtain  for  it  two  equal  values  with  contrary 


256  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

signs.  These  surfaces  are  therefore  divided  by  the  co-ordi- 
riate  planes  into  two  equal  and  symmetrical  parts.  The 
curves  in  which  these  planes  intersect  the  surfaces  are  called 
Principal  Sections,  and  the  axes  to  which  they  are  referred, 
Principal  Axes. 

If  now  the  surface  be  intersected  by  a  series  of  planes 
parallel  to  the  co-ordinate  planes,  the  intersections  will  be 
curves  of  the  second  order  referred  to  their  centre  and  axes, 
and  the  form  and  extent  of  these  intersections  will  determine 
the  character  of  the  surface  itself.  But  these  intersections 
will  evidently  depend  upon  the  signs  of  the  co-efficients  M, 
M',  M",  and  supposing  M  positive,  which  we  may  always  do, 
we  may  distinguish  the  following  cases : 

1st  case,  M'  and  M"  positive, 
2nd     "      M'  positive,  M"  negative, 
3d       "      M'  negative,  M"  positive, 
4th     "      M'  and  M"  negative. 

The  three  last  cases  always  give  two  co-efficients  of  the 
same  sign;  they  are  therefore  included  in  each -other,  and 
will  lead  to  the  same  results  by  changing  the  variables  in  the 
different  terms.  It  will  be  only  necessary  therefore  to  con 
sider  the  first  and  last  cases. 

CASE  I.  —  M,  M',  M",  being  positive. 
375.  Let  us  resume  the  equation 

M%2  +  My2  +  MV  +  L  =  o. 

Let  this  surface  be  intersected  by  planes  parallel  to  the 
co-ordinate  planes,  their  equations  will  be 

X  =  a,     y  =  /3,     2  =  7. 


CHAP.  TIL]  ANALYTICAL  GEOMETRY.  257 

Combining    these  with    the    equation  of  the  surface,  we 

have 

Mz2  +  My  +  MV  +  L  =  o, 
Mzs  +  M  V  +  M'/32  +  L  =  o, 
V  +  M/  +  L  =  o, 


for  the  equations  of  the  curves  of  intersection.  Comparing 
them  with  the  form  of  the  equation  of  the  ellipse,  we  see 
that  they  represent  ellipses  whose  centres  are  on  the  axes  of 
r,  y,  and  z. 

376.  To  determine  the  principal  sections,  make 

a  =  o,     /3  =  o,     7  =  0, 
and  their  equations  are 

Mz2  +  My  +  L  =  o, 

Mza  +  MV  +  L  =  o, 


which  also  represent  ellipses. 

377  If  L  =  o,  all  the  sections  as  well  as  the  surface  re- 
duce  to  a  point. 

If  L  be  positive,  the  sections  become  imaginary,  since 
their  equation  cannot  be  satisfied  for  any  real  values  of  the 
variables.  The  surface  is  therefore  imaginary. 

Finally,  ifL  be  negative,  and  equal  to  —  L',  the  sections 
will  be  real  so  long  as 

—  L'  +  MV,  —  L'  +  M'/32—  L'  +  M/, 

are  negative  ;  when  these  values  are  zero,  the  sections  and 
surface  reduce  to  a  point,  and  become  imaginary  for  all 
values  beyond  this  limit. 

This  surface  is  called  an  Ellipsoid. 
22*  2fl 


258 


ANALYTICAL  GEOMETRY. 


[CHAP.  VII. 


378.  If  we  make  y  =  o 
and  z  =  o  in  the  equa 
tion  of  the  ellipsoid,  the 
value  of  x  will  represent 
the  abscissa  of  the  points 
in  which  the  axis  of  x 
meets  the  surface.  We 
find 


The  double  sign  shows  that  there  are  two  points  of  inter 
sections,  symmetrically  situated  and  at  equal  distances  from 
the  origin. 

Making  in  the  same  manner  y  =  o,  and  x  =  o,  and  after 
wards  x  =  o  and  z  —  o,  we  obtain 


z  =  AB  = 


TJT  •        ?  =  AD  =  d=  \/ 


—  L 


M  V     M' 

The  double  of  these  values  are  the  axes  of  the  surface, 
and  we  see  that  they  can  only  be  real  when  L  is  negative. 

379.  The  equation  of  the  ellipsoid  takes  a  very  simple 
form  when  we  introduce  the  axes.  Representing  the  semi- 
axes  by  A,  B,  C,  we  have 

L  J±       r«_        — ' 

~  M7' '  ~  M7 '  ~  M ' 

and  substituting  the  values  of  M,  M',  M",  drawn  from  these 
equations  in  that  of  the  surface,  it  becomes 

A2BV  +  A2CV  4-  B2CV  =  A2B2C2. 


380.  If  we  make  the  cutting  planes  pass  through  the  axis 
of  z,  and  perpendicular  to  the  plane  of  xy,  their  equation 


CHAP.  VII.] 
will  be 


AXALYTICAL  GEOMETRY. 


Al 


or,  adopting  as  co-ordinates 
the  angle  NAC  =  9,  and  tne 
radius  AN=  r,  we  shall  have 

x  =  r  cos  9,     y  =  r  sin  9 ; 

and  substituting  these  values 
in  the  equation  of  the  sur 
face,  we  shall  have  for  the 
equation  of  the  intersection 
referred  to  the  co-ordinates  9,  z,  and  r, 

Mz2  +  r2  (M'  sin29  +  M"  cos29)  +  L  =  o. 

This  equation  will  represent  different  ellipses  according  to 
the  value  of  9.  If  M'  =  M",  the  axes  AC  and  AD  become 
equal,  the  angle  9  disappears,  and  we  have  simply 

Mz2  +  M'r2  +  L  =  o. 

Every  plane  passing  through  the  axis  of  z,  will  intersect 
the  surface  in  curves  which  will  be  equal  to  each  other,  and 
to  the  principal  sections  in  the  planes  of  xz  and  yz.  The 
third  principal  section  becomes  the  circumference  of  a  circle, 
and  all  the  sections  made  by  parallel  lines  will  also  be  circles, 
but  with  unequal  radii.  The  surface  may  therefore  be  gene 
rated  by  the  revolution  of  the  ellipse  BC  or  BD  around  the 
axis  of  z. 

This  surface  is  called  an  Ellipsoid  of  Revolution. 

381.  The  supposition  of  M  =  M',  or  M  =  M",  would  have 
given  an  ellipsoid  of  revolution  around  the  axes  of  x  and  y. 

882.  If  M  =  M'  =  M"  the  three  axes  A,  B,  C,  are  equal, 
and  the  equation  of  the  surface  becomes 


260  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

z2  +  y*  +  x2  +  ^  -  o, 

which  is  the  equation  of  a  Sphere. 

383.  Generally,  as  the  quantities  M,  M',  M",  diminish,  L 
remaining  constant,  the  axes  which  correspond  to  them  aug 
ment,  and  the  ellipsoid  is  elongated  in  the  direction  of  the 
axis  which  increases.     If  one  of  them,  as  M",  becomes  zero, 
the  corresponding  axis  becomes  infinite,  and  the  ellipsoid  is 
changed   into  a  cylinder,  whose   axis  is  the  axis  of  z,  and 
whose  equation  is 

Mz2  +  My2  +  L  =  o. 

The  base  of  this  cylinder  is  the  ellipse  BD.     (See  figure, 
Art.  378.) 

384.  If  M"  =  o,  and  M  =  M',  the  ellipse  BD  becomes  a 
circle,  and  the  cylinder  becomes  a  right  cylinder  with  a  cir 
cular  base.     This  is  the  cylinder  known  in  Geometry. 

385.  Finally,  if  M"  =  o,  and  M'  =  o,  the  equation  reduces 

to 

Mz2  +  L  =  o, 

which  gives 

z  =  d 


M 

This  equation  represents  two  planes,  parallel  to  that  of  xy 
and  at  equal  distances  above  and  below  it. 

CASE  II. — M  positive,  M'  and  M"  negative. 

386.  In  this  case  the  equation  of  the  surface  becomes 

Mz2  — My  — MV  +  L-o, 


CHAP.  TIL] 


ANALYTICAL  GEOMETRY. 


201 


and  the  equations  of  the  intersections  parallel  to  the  co-or 
dinate  planes  are 

Mz1  —  My  —  M'  a*  +  L  =  o, 
M^2  —  MV—  M/32  +  L  =  of 
My  +  MV—  My  +  L  =  o. 

The  two  first  represent  hyperbolas ;  the  last  is  an  ellipse. 
The  sections  parallel  to  the  planes  of  xz  and  yz  are  always 
real.  The  section  parallel  to  xy  will  be  always  real  \vhen 
L  is  positive.  If  L  be  negative  and  equal  to  —  L',  it  will 
be  imaginary  for  all  values  of  y,  which  make  the  quantity 
(L'  —  My)  positive  :  when  we  have  L'  —  My  =  o,  it  reduces 
to  a  point.  Thus,  in  these  two  cases,  the  surface  extends 
indefinitely  in  every  direction,  but  its  form  is  not  the  same. 

387.  Making  a  =  o,  /3  —  o,  y  —  o,  we  have  for  the  equa 
tions  of  the  principal  sections, 

Mz2  —  My  +L  =  o, 
M%2  —  M  V  +  L  =  o, 
My  +  MV— L  =  o. 

When  L  is  positive,  the 
two  first,  wThich  are  hyper 
bolas,  have  the  axis  of  z  for 
a  conjugate  axis,  and  are 
situated  as  in  the  figure. 
Every  plane  parallel  to  the 
plane  of  xy  produces  sections 
which  are  ellipses. 


262  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

388.  Making  two  of  the  co-ordinates  successively  equal  to 
zero,  we  may  find  the  expressions  for  the  semi-axes,  as  in  Art. 
348;  and  representing  them  respectively  by  A,  B,  C  V  —  i. 
and  introducing  them  in  the  equation  of  the  surface,  it 
becomes 


A2BY  —  A2Cy  — 


A2B2C2  -  o. 


(I) 


389.  When  L  is  negative,  the  principal 
sections,  which  are  hyperbolas,  have  BB'  for 
the  transverse  axis;  the  surface  is  imagi 
nary  from  B  to  B',  and  the  secant  planes 
between  these  limits  do  not  meet  the  sur 
face.  In  this  case,  the  semi-axes  will  be 
found  to  be  A  V — 1,  B  \/ — 1,  and  C, 
and  the  equation  of  the  surface  becomes 

A2BY  —  A2Cy  —  B2C  V  —  A2B2C2  =  o.  (2) 


The  surfaces  represented  by  equations  (1)  and  (2)  are 
called  Hyperboloids.  In  the  first,  two  of  the  axes  are  real, 
the  third  being  imaginary;  and  in  the  second,  two  are 
imaginary,  the  third  being  real. 

390.  If  M'  =  M",  we  have   A  =  B,  these  two    surfaces 
become  Hyperboloids  of  Revolution  about  the  axis  of  i. 

391.  If  M"  =  o,  the  corresponding   axis  becomes  infinite 
and    the    surface    becomes  a  cylinder  perpendicular  to   the 
plane  of  zy,  whose  base  is  a  hyperbola.     The  situation  of 
the  cylinder  depends  upon  the  sign  ot'  L.     Its  equation  is 


—  My  +  L  -  o. 


CHAP.  VII.]  ANALYTICAL  GEOMETRY.  2C3 

If  L  diminish,  positively  or  negatively,  the  interval  BB 
diminishes,  and  when  L  =  o,  we  have  BB'  =  o.  The  prin 
cipal  sections  in  the  planes  of  zx  and  yz  become  straight 
lines,  and  the  surfaces  reduce  to  a  right  cone  with  an  ellip 
tical  base,  having  its  vertex  at  the  origin  of  co-ordinates. 
In  this  case,  we  have  the  equation 

Mz2  —  My  —  MV  -  o. 

Sections  made  by  planes  parallel  to  the  planes  of  xz  and 
yz,  are  still  hyperbolas,  which  have  their  centre  on  the  axis 
of  y  or  x. 

392.  If  M"  =  o,  the  cone  reduces  to  two  planes  perpen 
dicular  to  the  planes  of  yz,  and  passing  through  the  ongin. 

393.  The  cone  which  we  have  just  considered,  is  to  the 
hyperboloids  what    asymptotes    are  to  hyperbolas,  and  the 
same   property  may  be  demonstrated    to    belong  to   them, 
which  has  been   discovered  in  Art.  242.     If  we  represent 
by  z  and  z',  the   respective   co-ordinates  of  the   cone   and 
hyperboloid,  we  shall  have 

My  +  M  V  My  +  M  V  —  L 

f       - — —     ..      .   .-  —    .  *t  *    — — 

which  gives 


_ 

M 


,_ 

" 


M  (z  +  z') 


The  sign  of  this  difference  will  depend  upon  that  of  L, 
hence,  the  cone  will  be  interior  to  the  hyperboloid,  when  L 
is  positive,  and  exterior  to  it,  when  L  is  negative.  The  dif 
ference  z  —  z  will  constantly  diminish,  as  z  and  z  increase, 
hence  the  cone  will  continually  approach  the  hyperboloid, 
without  ever  coinciding  with  it. 


264  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

Of  Surfaces  of  the  Second  Order  which  have  no  Centre. 

394.  Let  us  resume  the  equation 

M*2  +  My  +  M"*2  +  Hz  +  U'y  +  U"x  =  o.      (2) 

We  have  seen  (Art.  372),  that  this   equation  represents 
surfaces  which  have  no  centre  when  M,  M',  or  M"  is  zero. 
As  these  three  quantities  cannot  be  zero  at  the  same,  since 
the  equation  would  then  reduce  to  that  of  a  plane  (Art.  100) 
we  may  distinguish  two  cases ; 

1st  case,  M"  equal  to  zero. 

2d  case,  M"  and  M'  equal  to  zero. 

CASE  I. — M"  equal  to  zero. 

395.  The  above  equation  under  this  supposition  reduces  ic 

Mz2  +  My  +  Hz  +  ll'y  +  R"x  =  o. 

If  we  refer  this  equation  to  a  new  system  of  co-ordinates 
taken  parallel  to  the  old,  we  may  give  such  values  to  the 
independent  constants  as  to  cause  the  co-efficients  H'  and  H 
to  disappear,  (Art.  341).  The  equation  will  then  become 

M%2  +  My  +  H'x-o. 

396.  The  sections  parallel  to  the  co-ordinate  planes  are 

Mr2  +  U"x  +  M'/32  =  o, 
My  +  K";c  +  M72  =  o, 
Mz2  +  My  +  H"a  =  o. 

The  two  first  represent  parabolas,  and  are  always  real 
The  third  equation  will  represent  an  ellipse  or  hyperbola, 
according  to  the  sign  of  M  and  M'. 


CHAP.  VII.]  ANALYTICAL  GEOMETRY.  265 

397.  The  principal  sections  are 

Mr2  +  M  y  =  o,    Mr2  +  Wx  =  o,    My  +  H  "x  =  o. 

The  first  of  these  equations  will  represent  a  point,  or  two 
straight  lines,  according  to  the  sign  of  M'.  The  two  others 
represent  parabolas. 

398.  Let    us   suppose   M   and    M'   positive,  the   sections 
parallel  to  the  plane  of  yz,  and  whose  equation  is 

Ms2  +  My  +  H"a  =  o, 

will  only  be  real  when  H"  and  a  have  contrary  signs.  The 
surface,  therefore,  will  extend  indefinitely  on  the  positive 
side  of  the  plane  of  yz,  when  H"is  negative,  and  on  the 
negative  side  when  H"is  positive. 

399.  If  M'  be    negative,  the   equations  of  the    principal 
sections  are 


Mz2  —  M 


o, 


z2  +  Wx  =  o,     My  —  Wx  =  o. 


The  two  last  represent  parabolas,  having  their  branches 
extending   in   opposite   directions,  and    their  vertex  at  the 


266  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

origin  A.     The  sections  parallel  to  the  plane  of  yz,  will  be 
the  hyperbolas  B,  B',  B",  C,  C',  C". 

The    surfaces  which  we  have  just    discussed   are   called 
Paraboloids. 


CASE  II.  —  M'  and   M"  equal  to  zero. 

* 

400.  Equation  (2)  under  this  supposition  reduces  to 
Mz2  +  Hz  +  K'y  +  H"x  =  o. 

Moving  the  origin  of  co-ordinates  so  as  to  cause  tne  term 
Hz  to  disappear,  this  equation  becomes 

Mz2  +  H'y  +  H"  =  o. 
The  principal  sections  of  this  surface  are 

Mz2  +  H'y  =  o,     +  z2  +  R"x  =  o,    H'y  +  R"x  =  o. 
and  the  sections  parallel  to  the  co-ordinate  planes 

Mz2  +  H'?/  +  H"a  =  o, 
Mz2  +  H"*  +  H'/3  =  o, 
ll'y  +  Wx  +  M/  =  o. 

The  two  first  equations  of  the  parallel  sections  represent 
parabolas  which  are  equal  and  parallel  to  the  corresponding 
principal  sections.  The  sections  parallel  to  the  plane  of  xy 
are  two  straight  lines  parallel  to  each  other,  and  to  intersec 
tion  of  the  surface  by  this  plane.  The  surface  is,  therefore 


CHAP.  VII.]  ANALYTICAL  GEOMETRY.  267 

(hat  of  a  cylinder  with  a  parabolic  base,  whose  elements  are 
parallel  to  the  plane  of  xy.  The  projections  of  these  ele 
ments  on  the  plane  of  xy,  make  an  angle  with  the  axis  of  x 

TT// 

the  trigonometrical  tangent  of  wh4(^y  is  —  -y=-« 


Of  Tangent  Planes  to  Surfaces  of  the  Second  Order. 

401.  A  tangent  plane  to  a  curved  surface  at  any  point  is 
the  locus  of  all  lines  drawn  tangent  to  the  surface  at  this 
point. 

402.  Let  us  seek  the  equation  of  a  tangent  plane  to  sur 
faces  of  the  second  order.     Resuming  the  equation 

Az2  +  Ay  +  AV  +  Eyz  +  E'xz  +  E'xy  +  Cz  + 
C'y  +  C"x  +  F  =  o, 

and  transforming  it,  so  as  to  cause  the  terms  containing  the 
rectangle  of  the  variables  to  disappear,  we  have 

Az2  +  Ay  +  AV  +.  Cz  +  C'y  +  C"x  +  F  =  o.  (1) 

Let  x",  y",  z",  be  the  co-ordinates  of  the  point  of  tan- 
gency,  they  must  satisfy  the  equation  of  the  surface,  and  we 
have 

Az"2  +  A'?/"2  +  A' V2  +  Cz"  +  C'y"  +  CV  +  F  =  o. 

The  equations  of  any  straight  line  drawn  through  this 
point  are  (Art.  84), 

x  —  x"  =  a  (z  —  z"),     y  —  y"  =  b  (z  —  z"> 


268  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

For  the  points  in  which  this  line  meets  the  surface,  these 
equations  subsist  at  the  same  time  with  that  of  the  surface. 
Combining  them,  we  have 

A  (z  +  z")  (z  —  z")  +  A'(y  +  y")  (y  —  y")  +  A"  (x  +  x") 
(x—x")  +  C(z  —  z")  +  C'  (y  —  y")  +  C"  (x  —  a;")  =  o. 

Putting  for  y  —  y"  and  x  —  x",  their  values  drawn  from 
the  equations  of  the  straight  line,  we  have 

j  A  (z  +  z")  +  A'b  (y  +  y")  +  A"a  (x  +  x")  +  C  +  C'b  +  C"a\ 

(z  —  z")  =  o. 

This  equation  is  satisfied  when  z  —  z"  =  o,  which  gives 
z  =  z",  a:  =  x",  and  ?/  =  y".  Suppressing  (z  —  z"),  we 
have 

A  (z  +  z"^  J-  A'*  (y  +  y")  +  A" a  (x  +  a?")  +  C  +  C'b  +  C"a  =  o. 

This  equation  determines  the  co-ordinates  of  the  secona 
point  in  which  the  line  meets  the  surface.  But  if  this  line 
becomes  a  tangent,  the  co-ordinates  of  the  second  point  will 
be  the  same  as  those  of  the  point  of  tangency,  we  shall  have 
therefore 

a:  =  x",     y  =  y",     z  =  z", 
which  gives 

2Az"  +  2A'by"  +  ZA'ax"  +  C  +  C'b  +  C"a  =  o, 

for  the  condition  that  a  straight  line  be  tangent  to  a  surface 
vf  the  second  order.  Since  this  equation  does  not  determine 
the  two  quantities  a  and  b,  it  follows  that  an  infinite  number 
of  lines  may  be  drawn  tangent  to  this  surface  at  any  point. 
If  a  and  b  be  eliminated  by  means  of  their  values  taken  from 


CHAP.  VII.]  ANALYTICAL  GEOMETRY.  C69 

the  equations  of  the  straightjine,  the  resulting  equation  will 
be  that  of  the  locus  of  these  tangents.  The  elimination  gives 

(2Az"  +  C)  (z  —  z")  +  (2A'y"  +  C')  (y  —  y") 
+  (2AV  +  C")  (or  — a?")  =  o; 

and  since  this  equation  is  of  the  first  degree  with  respect  to 
x,  y,  and  z,  the  locus  of  these  tangents  is  a  plane  which  is 
itself  tangent  to  the  surface. 

403.  Developing  this   last   equation,  and   making  use  of 
equation  (1),  the  equation  of  the  tangent  plane  may  be  put 
under  the  form 

(2Az"  +  C)  z  +  (2A'y"  +  C')  y  +  (2A'V  +  C")  x 
+  Cz"  +  C'y"  +  C  V  +  2F  =  o. 

404.  For  surfaces  which  have  a  centre,  C,  C',  C",  are  zero, 
and  the  .equation  of  their  tangent  plane  becomes 

Azz"  +  A  yy"  +  A"xx"  +  F  ==  o. 


GENERAL   EXAMPLES. 

405.  "We  now  proceed  to  give  some  general  examples  upon 
Analytical  Geometry,  the  solution  of  which  will  prove  a 
valuable  exercise  for  the  student  in  familiarizing  him  with  the 
principles  of  the  science,  and  in  rendering  him  expert  in  their 
application.  The  co-ordinate  axes  are  supposed  rectangular, 
unless  the  contrary  is  indicated. 

1.  Find  the  equation  of  a  line  passing  through  a  given 
point  and  making  a  given  angle  with  the  axis  of  x. 

Find  the  equations  of  the  lines  which  shall  pass  through 
23* 


270  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

the  given  points  xf,  y'  ,  z1  ',  and  be  parallel  to  the  lines  whose 
equations  are  given. 


—  1  =  Ja:,  ty  +  z  =  2. 
Find  the  line  of  intersection  of  these  planes  : 

4 


Determine  the   points    of  intersection  of  these  lines   and 
planes : 

5.  •<  z  —  x  =  5  ; 

1 2x  —  \y  +  1  =  4z. 

{5x  —  4z  =  1, 
3#  =  2  —  82; 

{15#  —  2<?2  =  3, 
3y  =  1160  +  6 ; 

8.  Find  the  equation  of  the  line  passing  through  the  point 
x1  =  1,  yf  =  —  2,  3'  =  J,  and  parallel  to  the  plane  1J?/  —  9  = 

9.  Also  of  the  line  through  the  point  xr  =  — 8,  yf  =  —  1, 
2'  =  —  2J,  and  parallel  to  the  plane  42  +  3  —  x  =  Si/. 

10.  Find  the  equation  of  the    plane  passing  through  the 

three    points,    xl  =  J,   yi  =  —  1>    S]  =  2  ;    #2  =  3,    y2  ~  t> 
i  .   „         -t        __  ^i    ,.,  c 

11.  Do  the  lines  8x — 22=  1.  2y  —  J^  =  4,  and  Ja;  +  3=  62, 
Jg  4.  5  =  4y,  lie  in  the  same  plane  ? 


=2s  —  1,     ]  f  62;  +  13  =  82, 


CHAP.  TIL]  ANALYTICAL  GEOMETRY.  C71 

12.  Do  these,  x  —  z  =  1,  y  +  7*  =  3  ;  and  1  J.r  —  *z  =  2, 
5y  —  |  =  22? 

Find  the  equations  of  the  planes  containing  these  lines  : 
2*  +  7  =  3*,1    ig     r3*=2z-3, 
By  =  z  +  3  ;  j  }  4^  +  1   =  3*. 

f  62;  +  13  =  82, 
I8y  =  3z  —  14. 
1  5.  Find  the  equations  of  a  line  perpendicular  to  the  plane, 
&x-z  +  t  =  $y. 

16.  Find  thnt  of  a  plane  perpendicular  to  the  line  x  +  3  =• 
2z,  3-  —  4  =  24y. 

17.  A  plane  may  be  generated  by  a  right  line  moving  alo^g 
another  right  line  as  a  directrix,  and  continuing  parallel  to 
itself  in  all  its  positions  ;  find  the  equation  of  the  plane  from 
this  mode  of  generation. 

18.  Find  the  equation  of  a  line  passing  through  a  given 
point  in  a  plane,  and  making  a  given  angle  with  a  given  line  ; 
find  also  the  distance  from  the  given  point  to  the  point  of  in 
tersection  of  the  two  lines.    Discuss  the  result,  examining  the 
cases  in  which  the  given  angle  is,  0°,  45°,  and  90°. 

19.  Find  the  angle  included  between  a  line  and  plane  given 
by  their  equations.  —  This  problem  may  be  readily  solved  by 
means  of  the  following  considerations  :  the  angle  made  by  the 
line  and  plane,  is  that  included  between  the  line  and  its  projec 
tion  on  the  plane.     If  then,  a  perpendicular  to  the  plane  be 
drawn  from  any  point  on  the  line,  this  perpendicular,  with  a 
portion  of  the  given  line  and  its  projection  on  the  plane,  *ill 
form  a  right  angled  triangle,  of  which  the  angle  at  the  base  is 
the  one  sought.     The  angle  included  between  the  given  line 
and  the  perpendicular  is  the  complement  of  the  angle  at  the 
base,  and  may  be  readily  determined,  and  by  means  of  it  tue 


272  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

required   angle   is   instantly  found.     Denoting   the   required 
angle  by  V,  we  thus  find, 

Aa  -f  B6  +  C 

sin.  v^   ~~  — ~ — ~ "  —       •__      

Vl  -|-  az  +  tf  v/A2  -f-  BM^C2' 

"  20.  Find  the  angle  between  two  planes  given  by  their  equa 
tions.  —  If  from  a  point  within  the  angle  made  by  the  planes, 
we  draw  two  lines,  one  perpendicular  to  each  plane,  the  angle 
made  by  one  of  these  lines  with  the  prolongation  of  the  other, 
will  be  equal  to  the  angle  included  between  the  planes,  and 
may  be  easily  found.  Calling  the  required  angle  W,  we  thus 

obtain, 

AA'  +  BB'  +  CfC' 
cos  W  =  =t 


'  +  B2  +  C2  vA'2  +  B/2  +  C'2 
From  these  last  two  problems  we  can  easily  find  the  conditions 
for  parallelism  and  perpendicularity  between  a  line  and  plane, 
or  between  two  planes. 

21.  Find  the  equation  of  a  plane  passing  through  the  point 
xl  —  —  |,  yl  =  3,  zl  =  —  2,  and  perpendicular  to  the  plane 
3x  =  10  =  4y  +  z. 

22.  Show  that  the  three  lines  drawn  from  the  three  angles 
of  a  triangle  perpendicular  to  the  opposite  sides,  all  meet  at  a 
common  point. 

23.  Show  that  the  three  lines  drawn  from  the  three  angles 
of  a  triangle  to  the  middle  points  of  the  opposite  sides,  all 
meet  in  a  common  point. 

24.  Show  that  the  three  perpendiculars  erected  upon  the 
sides  of  a  triangle  at  their  middle  points,  all  meet  in  a  common 
point. 

25.  Having  given  a  point  in  space,  and  a  plane,  find  th<; 
shortest  distance  from  the  point  to  the  plane.     If  the  co-ordi 
nates  of  the  given  point  be  designated  by  x',  y',  z',  and  the 


CHAP.  TIL]  ANALYTICAL  GEOMETRY.  273 

equation  of  the  plane  be,  z  =  Ax  -f  By  -f  D,   the    required 
distance  is, 

D  +  Ax'  +  By'  —  z> 
v'l  +  A2  4-  B2 

26.  Find  the  equation  of  a  line  tangent  to  a  circle  and 
parallel  to  a  given  line. 

27.  Find  the  equation  of  the  tangent  line  to  the  circle  by 
means  of  the  property  that  this  tangent  is  perpendicular  to 
the  radius  through  the  point  of  contact. 

28.  Find   the   equation   of  a  tangent   line   to   the   circle, 
(*/-6)2  +  (z.-a)2=R2. 

Ans.  (y  —  b)  (y"  —  b)  +  (x  —  a)  (x"  —  a)  =  R2. 

(Henceforth,  in  designating  points  in  a  plane,  we  shall 
simply  give  the  values  of  the  co-ordinates  in  the  order,  x,  y  ; 
thus,  the  point  (2,  5),  would  signify  the  point  whose  co-ordi 
nates  are  x  =  2,  y  =  5.  For  points  in  space  the  co-ordinates 
will  be  given  in  the  order,  x,  y,  z.) 

29.  Find  the  equation  of  the  tangent  line  to  the  ellipse, 
9y*  -f  fy2  =  144,  at  the  point  (3,  3) ;  also  that  of  the  normal 
at  the  same  point ;  likewise  the  lengths  of  the  subtangent  and 
subnormal  on  both  axes. 

30.  Find  the  equation   of  the  tangent  line  to  the  ellipse 
parallel  to  a  given  line ;  also  that  of  the  normal  subjected  to 
a  similar  condition. 

31.  Find  the  equation  of  the  ellipse,  which,  with  a  trans 
verse  axis  equal  to  18,  shall  pass  through  the  point  (6,  7). 

32.  Find   that   of  the    ellipse  which   passing  through  tht 
point  (5J,  8),  shall  have  its  conjugate  axis  equal  to  10. 

33.  Determine  the  area  of  the  ellipse,  16^  -f  13z2  =  182. 

2K 


274  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

34.  Arc  the  lines  y  =  2x —  3,  y  ^=  3x  —  6,  supplemental 
chords  of  the  ellipse  9/  -f  lx*  =  144  ? 

35.  The  equation  of  one  supplementary  chord  in  the  ellipse 
9?/2  -f  4z2  =  36,  is  2y  =  x  +  3 ;  find  that  of  the  other. 

36.  Are  the  lines  3?/  =  5#,  2J?/  =  4x,  conjugate  diameters 
of  the  ellipse  Sf  -f  5z2  =  30  ? 

37.  In  the  ellipse  10 if  +  Qxz  =  42,  find  the  equation  of  that 
diameter  which  is  the  conjugate  of  the  one  whose  equation  is, 
6y  =  Tar. 

38.  In  the  ellipse,  it  is  often  desirahle  to  know  that  pair  of 
conjugate  diameters  whose  lengths  are  equal.     For  this  pur 
pose  take  the  value  of  A'2,  and  the  second  value  of  B/2  (Art. 

:  185)  and  place  them  equal  to  each   other.     We    shall   thus 
obtain,  A2B2  -f  A2B2  tang2  cc  =  A4  tang2  cc  +  B4,  which  gives 

-n 

tang  oc  =  ±  -r ,  henpe,  the  required  diameters  are  parallel  to 

the  chords  joining  the  extremities  of  the  axis. 

39.  Show  that  the  angles  included  by  these  equal  conjugate 
diameters,  are  the  greatest  and  smallest  which  can  be  con 
tained  by  any  pair  of  conjugate  diameters  of  an  ellipse,  and 
consequently  constitute  the  limits  alluded  to  in  Art.  188. 

40.  Show  that  in  the  ellipse  the  curve  is  cut  by  loth  the 
diameters  conjugate  with  each  other. 

41.  Show  that  in  the  hyperbola,  the  curve  can  be  cut  by 
only  one  of  two  conjugate  diameters. 

42.  The  lines  2y  =  x  +  12,  8y  +  x  =  12,  are  supplemental 
chords  of  an  ellipse  whose  transverse  axis  is  24 ;  what  is  the 
equation  of  the  curve  ?  Ans.  16/  -f  z2  =  144. 

43.  Find  the  equation  of  the  tangent  line  to  the  parabola 
?/2  =  4x,  at  the  point  (4,  4) ;  also  that  of  the  normal :  and 


CHAP.  VII.]  ANALYTICAL  GEOMETRY.  275 

that  of  a  line  through  the  focus  and  point  of  tangency ;  and 
find  the  angle  included  between  this  last  line  and  the 
tangent. 

44.  Find  the  equation  of  a  line  which  shall  be  tangent  to 
the  parabola  y-  =  8x,  and  parallel  to  the  line  y  +  1  =  ox. 

45.  Find  the  equation  of  the  parabola,  which  with  a  para 
meter  equal  to  12,  shall  pass  through  the  point  (2,  8). 

46.  What  is  the  area  of  the  segment  cut  off  from  the  para 
bola  3#2  =  &2x,  by  the  line  y  =  '2x  —  4  ?  Ans.  18;,;. 

47.  What  is  the  area  of  the  segment  cut  off  from  the  para 
bola  8/  =  2312:  —  724,  by  the  line  8y  =  llx  —  4  ? 

Ans.  7; 

48.  In  the  hyperbola  9/  —  4z2  =  —  36,  find  the  equation 
of  the  diameter  which  is  the  conjugate  of  the  one,  y  =  2x. 

49.  In  the  same  hyperbola,  are   the  lines  2#  =  #,  y  =  os. 
conjugate  diameters  ?  9 

50.  Are  these,  5#  =  2z,  and  9#  =  lOx  f 

51.  Are  the  lines  -y  =  5x,  4?/  =  x,  conjugate  diameters  of 
the  circle  x1  -f  y"  =  14  ? 

52.  Are  the  lines  y  =  3.r,  y  =  4z,  conjugate  diameters  of 
the  ellipse  10/  -f  8^2  =  40  ? 

53.  Find  the  equation    of   the    ellipse  for  which  they  are 
conjugate  diameters :  also  the  equation  of  the  curve  referred 
to  them. 

54.  Find   the    equation  of  the   hyperbola  which,  with  its 
transverse  axis  equal  to  16,  has  the  lines  3j/  =  2x,  3j/  +  2x=  0, 
for  its  asymptotes. 

55.  Find  the  equation  of  a  hyperbola  passing  through  the 
point  (1,  2),  and  having  one  of  its  asymptotes  parallel  to  the 
line,  3#  =  2x  +  3.  Ans.  4z2  —  9^/2  =  —  32. 

56.  From  the  equation,  b  sin  cc1  — p  cos  a1  =  0,  (Art.  213), 


276  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

P  %P 

we  obtain,  sin2  a1  =  9  ;  and  the  parameter  2pl  =  -^-— , 

•^t*     f     /^  8111.     OC 

then  becomes,  2pl  =  2(p  +  2a)  =  4FM  (see  figure  to  Art. 
212).  Hence,  In  the  parabola,  the  parameter  of  any  diameter 
is  four  times  the  distance  of  its  vertex  from  the  focus. 

57.  In  the  parabola  y2  =  8x,  what  is  the  parameter  of  the 
diameter,  y  =  16  ?  Ans.  136. 

•r>8.  Show  how  you  may,  from  Arts.  215,  216,  derive  a 
shnpk  graphic  construction,  for  drawing  a  line  tangent  to  a 
j  irabola  and  parallel  to  a  given  line. 

59.  Demonstrate  generally,  that  in  an</  conic  section  the 
chords  bisected  by  a  diameter  are  parallel  to  the  tangent  at  the 
extremity  of  that  diameter. 

60.  Find  the  equation  of  a  tangent  plane  to  the  sphere 
(x  —  «)2  +  (y—  5)2  -f  (z  —  c?  =  R2  at  the  point  (x»  y»  z]?)  by 
means  of  the  property  that  this  tangent  plane  is  perpendicular 
to  the  radius  through  the  point  of  contact. 

Ans.  (x—a)(x}  —  a)  +  (1/  —  b)^]  —  b)  +  (z  —  c)(z1  —  c)  =  'R2. 

61.  Given  the  base  of  a  triangle  and  the  sum  of  the  tangents 
of  the  angles  at  the  base,  to  find  the  locus  of  the  vertex. 

Ans.  A  parabola. 

62.  Given  the  base  of  a  triangle  and  the  difference  of  the 
angles  at  the  base,  to  find  the  locus  of  the  vertex. 

Ans.  An  equilateral  hyperbola. 

63.  Required  the  locus  of  a  point  P,  from  which,  drawing 
perpendiculars  to  two  given  lines,  the  enclosed  quadrilateral 

shall  be  equivalent  to  a  given  square. 

Ans.  A  hyperbola. 

54.  Find  the  locus  of  the  intersections  of  tangent  lines  to 
the  parabola  with  perpendiculars  to  them  from  the  vertex. 

Ans.  A  cissoid. 


AXALYTICAL  GEOMETRY. 


277 


Fig.  24. 


CHAP.  TIL] 

65.  A  common  carpenter's  square  CBP  (Fig.  24),  moves 
so  that  the  ends  C  and  B  of  one  ^f  its 

sides,  remain  constantly  upon  the  two 
sides,  AX  and  AY,  of  the  right  angle 
TAX.  Required  the  curve  traced  by 
the  other  extremity  P. 

Ans.  An  ellipse. 

66.  Find  the  locus  of  the  vertex  of  a  parabola  which,  with 
a  given  focus,  is  tangent  to  a  given  line.  Ans.  A  circle. 

67.  Chords  are  drawn  from  the  vertex  of  a  conic  section  to 
points    of  the    curve.     Required  the   locus    of   their  middle 
points. 

68.  Given  the  base  and  altitude  of  a  triangle,  to  find  the 
locus  of  the  intersections  of  per 
pendiculars  from  the  angles  upon 

the  opposite  sides. 

Ans.  A  parabola. 

69.  Find  the  equation  of  the 
surface    generated   by   the    line 
BC  (Fig.  25)  moving  parallel  to 

the  plane  of  yz,  and  constantly  piercing  the  planes  of  xz,  and 
xy  in  the  given  lines  ZX,  ?/D,  the  last  line  being  parallel  to  AX. 

70.  Upon  the  plane  AC  (Fig. 
26)  inclined  at  an  angle  of  10° 
to  the  plane  AB  of  the  horizon, 
is  erected  a  pole,  HD,  perpen 
dicular  to  the  plane  AB :  over 
the  top  of  this  pole  is  stretched 

a  rope,  CHE,  whose  entire  length  is  150  feet,  its  extremities, 
E  and  C,  meeting  the  plane  AC  at  distances  DE,  and  DC, 
24 


_/'/>.  26. 


278  ANALYTICAL  GEOMETRY.  [CHAP.  VH. 

from  the  foot  of  tho  pole,  equal  each  to  12  feet.     Re-quired 
the  height  of  the  pole  DII.  Am.  74--J1G  feet,  nearly. 

71.  Find  the  equation  of  the  parabola  from  the  property 
exhibited  in  Art.  211. 

72.  Show  how  to  describe  a  parabola  when  you  have  given 
its  vertex  and  axis,  and  the  co-ordinates  of  ono  of  its  points. 

73.  Show  that   in  the    hyperbola,  the   tanyent    line  to  the 
curve  bisects  the  angle  included  between  the  two  linen  from  the 
foci  to  the  point  of  tanyency. 

74.  Show  how  you  may,  from  the  preceding  property,  dniw 
a  tangent  line  to  the  hyperbola  from  a  point  either  without  or 
upon  the  curve,  by  a  method  analogous  to  that  given  for  the 
ellipse  in  Art.  175. 

75.  Take  two  lines  not  in  the  same  plane,  and  pans  a  plane 
through  each.     Required  the  locus  of  the  line  of  intersection 
of  these  planes  when  they  are  subjected  to  the  condition  of 
continuing  perpendicular  to  each  other. 

An*.  A  hyperbolic  paraboloid. 

76.  In  the  ellipse  %2  -f  Gz2  =  48,  find  the  equation  of  the 
diameter  conjugate  with  the  one  whose  equation  is  Zy  =  7#, 
and  also  the  equation  of  the  curve  referred  to  tho.se  diameters. 

77.  What  is  the  equation  of  tho  hyperbola  of  which  the 
lines  by  =  2x9  y  =  4 Jar,  are  conjugate  diameters  ? 

Construct  the  following  curves,  and  also  the  asymptotes  and 
centres  of  such  as  have  them. 

78.  ty  —  4Xy  +  x>—y  +  ,^  =  10. 

79.  %2  —  2xy  —  x*  -f  y  —  6  =  x  —  20. 

80.  y'1  —  Qxy  -f  9^2  —  6 y  -f  &r  +  &  =  0 

81.  Kxy  -f  y?  —  y  -f  \x  =  J  —  f. 

82.  'i  —  x>  —     +  bx  =  6. 


OHAP,  \  ANALYTICAL  GKOM1T1  279 

-*y  +  I*  +  G  =  O. 

84.  4^/1  -f  ry  —  6zl  -f  2y  —  a:  =  1  .'. 

.  y*  +  2xy  -f  x1  —  Oy  — 6*  +  9  =  0. 

80.  Ly  4-  10./7/  -f  :J~V  +  8y  —  13*  +  24-0. 

ST.  .y2  —  r/y  —  :}0^  —  4y  -f  .00^  —  21  = 

88.  if  -f  1  \xy  -f  3^  —  4y  -  l\x  - 

89.  t/z  -  |-  :V  —  2y  —  lOa?  -f  19  -  0. 
'.».  L'V  -ar+1-3^  — 5. 

91.  4^  +  4^  +  ^  — 4y  —  8a?+ 16-0. 
'-'.   ^     •    I-./  ~2z*  +  C>y  —  4Qx=  .1. 


•'*•  ;/  + 

98.  y*  —  'Ixij  -f  **  -f-  4y  —  4*  -f-  3  «  0. 

99.  y*  +  2r y  -f  **  —  1 0  =5  0. 

1 00.  5yv  —  Or//  -f  2**  -f  y *  —  1. 

101.  -if  —  1-j-ij  -}-  1  Oy*  _  -  2//  -f  1  ,'Jz  t 

n\'f-  -)::,1]  now  give  Borno  rerj  useful  graphic 
relating  to  conic  section*,  leaving    I|K-  .-itiouH    HH 

-<  -  f'-r  iljc«  ^tu(]cnt.J 

102.  Having  given  a  pair    /<f.    ,,„          ^ 
of   conjugate    dknaetern,    HO  / 

and  BC  (Fig.  27),  of  ai,  7/  "*/  9/j"c 

tbe  curve  may  be  traced  by 
point*,  thua:  on  AC,  AJJ,  de 
scribe  the  parallelogram  Al>. 
Divide-  l)<;  into  May  number  of  «»jual  part»,  ;ifi«l  A^'  into  i  h  • 

nurjjber  of  j>art«,  albo  c'fjuaJ.     J>ruw   the  Jin*.-*   JJ1, 


280  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 

etc.,  from  B  to  the  points  of  division  on  DC  ;  and  the  lines 
El,  E2,  etc.,  from  E  to  the  points  of  division  on  AC.  The 
points  of  intersection  of  the  corresponding  lines  will  be  points 
of  the  curve. 

103.  The  following  is  a  good  method  for  describing  the 
ellipse  by  points,  when  we  have  given  a  pair  of  conjugate 
diameters.  Let  AC  (Fig.  28)  be  a  diameter,  and  AB  equal 


JJ 


SEv, 

*{ 

2/    N 


'.  28 


•JB      4:        &       *       *     J? 

and  parallel  to  its  conjugate.  Through  B  draw  BE  parallel 
to  AC  :  take  BE  any  multiple  of  AC :  produce  BA  and  take 
AD  the  same  multiple  of  AB :  divide  BE  into  any  number  of 
equal  parts,  and  AD  into  the  same  number  of  equal  parts : 
through  A  draw  lines  to  the  points  of  division  in  BE,  and 
from  C,  lines  to  those  in  AD.  The  intersections  of  the  corre 
sponding  lines  will  be  points  of  the  ellipse.  If  BE  be  taken 
to  the  right  of  B,  instead  of  to  the  left,  the  points  found  will 
belong  to  a  hyperbola. 

104.  Having  given,  in  length  and  position,  a  pair  of  con 
jugate  diameters  of  an  ellipse,  to  construct  the  axes.  Let 
AAr,  and  BB'  (Fig.  29),  be  the  given  conjugate  diameters. 
Through  A  draw  IAE'  perpendicular  to  OB,  and  on  this  line 
lay  off  on  each  side  of  A,  the  distances  AE,  AE',  each  equal 


AXALYTICAL  GEOMETRY. 


281 


CHAP.  VII.] 

to  the  semi-conjugate  diameter  OB.  Through  the  points  E, 
and  E',  thus  determined,  draw  from  the  centre  0  the  lines 
OE,  OE'.  Then  the  lines  D'OD,  HOH',  bisecting  the  angle 
EOE'  and  its  supplement, 
will  give  the  directions 
of  the  axes;  the  trans 
verse  axis  being  always 
situated  in  the  acute  angle 
formed  by  the  conjugate 
diameters.  The  length 
DD'  of  the  transverse  axis  is  given  by  the  sum  of  the  lines 
OE,  OE' ;  that  of  the  conjugate  axis  HH',  is  equal  to  their 
difference,  OE' —  OE.  This  construction  is  readily  demon 
strated  by  showing  that  the  loci  of  the  points  E,  E',  are  two 
circumferences  of  circles  concentric  with  the  ellipse,  and 
having  for  radii  (A  — B),  and  (A  +  B),  respectively;  and 
then  showing  that  the  lines  OE,  OE'  are  diameters  of  the 
curve  making  equal  angles  with  its  axis. 

105.  Let  AA'  and  BB'  (Fig.  30)  be  the  axes  of  an  ellipse. 
Take  a  ruler  Pm,  equal  in  length  to  the  semi-transverse  axis ; 
from    the    extremity   P,    lay    off 
PH  =    the   semi-conjugate   axis; 
now  move   this  ruler  so  that  the 
extremity  m  shall  remain  on  the 
conjugate    axis    BB',    while    the 
point  of  division  H  continues  upon 
the  transverse  axis  AA' :  then  the 

point  P  will  describe  the  ellipse.  This  principle  has  been 
applied  to  the  construction  of  a  very  simple  instrument  for 
describing  ellipses,  known  as  the  elliptic  compasses,  or 
trammels. 

24*  2L 


Jff 

/P 

£• 

/ 

/ 

^ 

o 

/JI 

J. 

7** 

Wg.  30. 

282  ANALYTICAL  GEOMETRY.  [CHAP.  VII, 

106.  Find  the  equation  of  the  right  line  referred  to  oblique 
axes  in  its  own  plane,  when  its  position  is  fixed  by  the  length 
and  direction  of  the  perpendicular  to  it  from  the  origin. 

An  s.  p  =  x  cos  cc  +  y  cos  /3,  when  ex  and  fi  are  the  angles 
made  by  the  perpendicular,  with  the  axes  of  x  and  y 
respectively.  If  the  axes  are  rectangular  the  equation  is, 
p  =  x  cos  cc  +  y  sin  ex. 

107.  Find  and  discuss  the  polar  equation  of  the  right  line. 

108.  Find  the  locus  of  the  centre  of  a  circle  inscribed  in  a 
sector  of  a  given  circle,  one  of  the  bounding  radii  of  the  sector 
being  fixed. 

109.  Show  that,  of  all  systems  of  conjugate  diameters  in 
an  ellipse,  the  axes  are  those  whose  sum  is  the  least,  while  the 
equal  conjugate  diameters  are  those  whose  sum  is  the  greatest. 

•110.  Find  the  locus  of  a  point  so  situated  upon  the  focal 
radius  vector  of  a  parabola,  that  its  distance  from  the  focus 
shall  be  equal  to  the  perpendicular  from  the  focus  to  the 
tangent.  Ans.  r  =  a  sec  Jd,  counting  6  from  the  vertex. 

111.  Show  that,  the  equation  of  the  tangent  line  to  the 
ellipse  referred  to  its  centre  and  axes,  may  be  put  under  the 
form 


while  that  for  the  hyperbola  may  be  written, 


y  =  mx  +  N/A2w2  —  B2  : 
and  that  of  the  parabola  is, 


These  equations  are  known  as  the  magical  equations  of  the 
tangent. 

112.  In  the  focal  distance  FP  of  any  point  P  of  a  parabola, 


CHAP.  TIL]  AXALYTICAL  GEOMETRY.  £83 

J?p  is  taken  equal  to  the  distance  of  P  from  the  axis ;  find 
the  locus  of  p. 

* 

Ans.  r  =  c  tang  J0,  estimating  6  towards  the  vertex. 

113.  Prove  that  the  right  lines  drawn  from  any  point  in  an 
equilateral  hyperbola  to  the  extremities  of  a  diameter,  make 
equal  angles  with  the  asymptotes. 

114.  Show  that  the  equation  of  the  plane  may  be  put  under 
the.  form, 

p  =  x  cos  O,  x)  +  y  cos  (p,  y)  +  z  cos  (p,  z), 
when  p  is  the  length  of  the  perpendicular  to  the  plane  from 
the  origin,  and  the  notation 

cos  (p,  x\  cos  (p,  y),  cos  (p,  z), 

is  use,d  to  signify  the  cosines  of  the  angles  made  by  this  per 
pendicular  with  the  axes  of  x,  y,  z,  respectively.  Or,  it  may 
written, 

p  =  x  sin  (P,  x)  +  y  sin  (P,  y)  +  z  sin  (P,  z), 
where 

sin  (P,  x\  sin  (P,  y\  sin  (P,  z), 

signify  the  sines  of  the  angles  made  by  the  plane  with  the 
axes  2-,  y,  2.  Using  an  analogous  notation  to  express  the 
angles  made  by  the  plane  with  the  co-ordinate  planes,  its 
equation  may  be  written, 

p  =  x  cos  (P,  yz)  +  y  cos  (P,  xz)  +  z  cos  (P,  xy\ 

Construction  of  Surfaces  of  the  Second  Order  from  their 
Equations. 

406.  This  consists  in  constructing,  from  the  equation  of  the 
surface,  its  principal  sections,  and  its  projections,  and  in  de 
termining  the  kind  of  the  surface.  Let  the  general  equation 
of  these  surfaces  be  solved  with  reference  to  z,  and  we  shall 
obtain, 


284  ANALYTICAL  GEOMETRY.  [CHAP.  VII. 


Writing  z  equal  to  the  rational  part  of  its  Value,  we  have, 


_ 

~2A 


which  represents  a  plane,  above  and  below  which  must  be  laid 

1          ... 
off  ordinates  equal  to  ?nr  v/<P  (#,  «/)>  in  order  to  obtain  points 

of  the  surface.  This  plane,  (N),  is  called  a  diametral  plane, 
since  it  bisects  a  system  of  parallel  chords  of  the  surface,  and 
passes  through  its  centre.  Similar  results  would  ensue  from 
solving  the  general  equation  with  reference  to  each  of  the  other 
variables  x9  and  y  ;  and  thus  we  should  obtain  three  of  these 
diametral  planes,  which,  intersecting  at  the  centre  of  the  sur 
face,  would  enable  us  to  determine  and  construct  that  point. 
Taking  the  radical  part  of  the  value  of  2,  and  placing  it  =  0, 
we  have,  <p  (x,  y)  =*  0,  which  manifestly  represents  the  projec 
tion  of  the  surface  upon  the  plane  of  xy*  Similarly,  we  may 
obtain  its  projection  on  xz  and  yz.  These  projections  being 
always  conic  sections,  may  be  readily  constructed. 

To  enter  into  a  full  exposition  of  the  process  for  determining 
the  species  of  the  surface,  would  involve  us  in  much  unneces 
sary  detail  and  repetition  of  principles  previously  discussed, 
besides  occupying  more  space  than  we  could  afford  to  it  in  the 
present  volume.  By  the  aid  of  the  principles  already  estab 
lished  and  the  examples  of  their  application  exhibited  in  the 
methods  of  discussing  curves  and  surfaces,  the  student  ought 
to  be  able,  with  a  moderate  degree  of  ingenuity,  to  effect  this 
investigation  for  himself.  He  will  experience  but  little  difficulty 
in  eliminating  the  necessary  analytical  criteria  for  determining 
the  species  of  anj  surface  of  the  second  order,  if  he  will  only 


CHAP.  VII.]  ANALYTICAL  GEOMETRY.  285 

keep  in  mind  the  mode  in  which  we  accomplished  the  same 
analysis  in  the  case  of  the  general  equation  of  the  second  degree 
between  two  variables.  We  subjoin  a  few  examples  for 
practice. 

115.  422  —  4xy  4  4/  +  5z2  —  32z  —  24z  -f  96  =  0. 

116.  x-+  ty  +  2z'+2xy  —  2x  —  4y—42  =  0. 

117.  x-  4  y1  4  2z-  —  Ixy  —  2xz  4  lyz  4  2y  —  3  *=  0. 

118.  or  —  2/  4  z2  +  -2xy  — >  4.T2  4  4y  +  4z  —  9  =  0. 

119.  3.r2  4  2/  —  2ar«  +  4^  —  4x  —  8z  —  8  =  0. 

120.  x2  +y2  +  2z*+  2xy  +  2xz  +  '2yz  —  2x—2y  +  2.2  =  0. 

121.  a;2  —  t/2  —  2ss  -f  2^  —  4yz  +  2y  +  2z  =  0. 

122.  rr2  +  3/  +  2^2  +  Ixy  +  4^  —  82:  —  4y  —  3^  =  0. 

123.  rcs  +^2  — 2z2  +  2^  -f  2xz  +  2^2  — 4a;  — 2y  +  2z  =  0. 

124.  2:2  -f  y-  +  922  —  2.ry  —  6r^/  +  6^/2  +  2z  —  4^;  =  0. 
Find  the  equations  of,  and  construct  the  planes  tangent  to 

these  surfaces^  at  the  points  given  : 

125.  \x-  —  8  (y2  +  z2)  4-  100  =  0,  at  (1,  2,  3). 

126.  ox-  +  6/  4  z-  —  30  =  0.  at  (1,  2,  1). 

127.  4.s2  —  Qy  —  Six  =  0,  at  (1,  3,  5). 

128.  8/  —  5z2  +  24x  =  0,  at  (J,  1,  2). 

129.  Find  the  equation  of  a  cone  having  its  vertex  on  the 
axis  of  z  at  a  distance  5  from  the  origin,  its  base  being  a 
hyperbola  in  the  plane  xy,  the  axes  of  this  hyperbola  being 
coincident  with  those  of  x  and  ?/,  their  numerical  values  being 
8  and  6.     Then  intersect  this  cone  by  a  plane  through  the 
axis  of  y  making  an  angle  of  45°  with  xy  and  find  the  equa 
tion  of  the  curve  of  intersection  of  the  plane  and  cone,  referred 
to  axes  in  its  own  plane,  and  construct  it. 

130.  Discuss  and  determine  the  form  of  the  surface  defined  by 
the  equation  aV  +  ?/V  —  rV  =  0  ;  show  how  it  may  be  gene 
rated,  and  then  find  its  equation  from  its  mode  of  generation. 


286  ANALYTICAL  GEOMETRY.  [CHAP.  TIL 

Ans.  It  is  a  conoid,  having  for  a  plane  director  the  plane 
xz,  and  for  directrices  the  axis  of  y  and  a  circle  x--{-y'i=r'1^ 
at  a  distance  a  from  the  origin. 

131.  CP,  CD,  are  conjugate  semi-diameters  of  an  ellipse: 
prove  that  the  sum  of  the  squares  of  the  distances  of  P,  D, 
from  a  fixed  diameter  is  invariable. 

132.  Show   that    the   equation    ?/2  —  'Zxy  sec   cc  -f  x*  =  0, 
represents  two  right  lines  passing  through  the  origin  and  in- 

cc 


(c\ 
45°  rh  27. 

133.  Determine  the  surface    represented  by  the    equation 
z  =  xy. 

134.  Show  that  if  at  any  point  of  a  hyperbola  a  tangent 
be  drawn,  the  portion  of  this  tangent  included  between  the 
asymptotes  will  be  equal  in  length  to  that    diameter  which 
is  the  conjugate  of   the  one    passing  through   the  point    of 
contact. 

135.  Find  the  equation  of  the  parabola  in  terms    of  the 
focal  radius  vector  and  the  perpendicular  from  the  focus  on 
the  tangent.         Ans.  d*  ==  %pr,  where  d  is  the  perpendicular. 

136.  Find  the  equations  of  the  sides  of  the  regular  hexagon 
inscribed  in  the  circle  x"1  +  ?/2  =  4. 

137.  Show  that,  if  at  the  extremity  of  the  ordinate  passing 
through  either  focus  of  the  ellipse  a  tangent  to  the  curve  be 
drawn,  and  at  the  point  in  which  this  tangent  meets  the  trans 
verse  axis  produced,  a  perpendicular  be  drawn  to  this  axis, 
then  the  ratio  of  the  distances  of  any  point  of  the  curve  from 
the  focus  and  this  line  is  constant  and  equal  to  the  eccentricity. 
These  lines  are  called  the  directrices  of  the  curve.     The  same 
property  belongs  to  the  hyperbola  also. 

138.  In  the  hyperbola,  16?/2  —  9z8  =  —  144,  find  the  equa- 


CHAP.  TIL]  ANALYTICAL  GEOMETRY.  287 

tion  of  the  diameter  conjugate  to  the  one,  2?/  =  rr,  and  find 
the  equation  of  the  curve  referred  to  these  diameters. 

139.  Find  the  equations  of  the  ellipse  and  hyperbola  referred 
to  the  focal  radius  vector  and  perpendicular  on  the  tangent. 

BV  BV 

Ans.  Ellipse,  f  =        zr     ;  Hyperbola,  f  = 


140.  Find  the  equations  of  the  same  curves  referred  to  the 
central  radius  vector  and  perpendicular  on  the  tangent. 

A2B2  A2BJ 

Ans.  Ellipse,  ^  =  A,  +  B2_p2;  Hyperbola,  /=y_A*  +  B" 


NOTES. 


I. — Art.  150,  p.  107.  In  the  discussion  of  the  equation  at  the  bottom  of 
this  page,  the  positive  abscissas  must  be  reckoned  to  the  left,  and  the  negative 
abscissas  to  the  right.  This  results  from  the  nature  of  the  transformation  em 
ployed  in  this  article  for  removing  the  origin  from  0  to  B.  The  formula  used  for 
this  purpose  is  x  =  OB  —  a/,  where  x  and  x/  having  contrary  signs  must  be 
reckoned  in  contrary  directions,  and  since  the  positive  values  of  x  were  counted 
to  the  right,  those  of  x'  must,  in  the  transformed  equation,  be  counted  to  the 
left.  This  becomes  more  apparent  by  referring  to  Art.  110,  where  we  found 
the  formula  for  passing  from  one  set  of  co-ordinates  to  a  parallel  set,  to  be, 
x  •=  a  -f-  z',  where  the  positive  values  of  both  x  and  x'  are  counted  in  the 
same  direction,  and  so  these  quantities  have  like  signs  in  the  formula.  But 
had  the  positive  values  of  a/  been  reckoned  in  a  contrary  direction  to  that  in 
which  we  estimated  those  of  z,  then  the  formula  would  have  been  x  =  a  —  x', 
the  change  of  direction  in  zr  being  indicated  by  its  change  of  sign.  When 
the  origin  is  removed  from  B  to  A  (page  109),  the  direction  of  the  positive 
abscissas  is  again  reversed  by  the  formula  employed,  and  in  the  resulting 
equation  they  must  be  reckoned  to  the  right. 

II.  — Arts.  224-5.  The  same  remark  holds  good  here,  the  origin  being  at 
W,  and  the  negative  abscissas  counted  to  the  right,  that  is,  from  B'  towards 
B.  In  Art.  226,  where  the  origin  is  transferred  from  W  to  A,  the  formula 

c  sin  v  cos  v  cos  u 

should  be,  •£  —  — : — : ;  —  *»  by  which,  since  x  and  a/  have 

sin  (v  -f-  M)  sin  (v  —  if) 

contrary  signs,  the  direction  of  the  positive  abscissas  is  again  reversed,  and 
must,  in  the  resulting  equation,  be  counted  to  the  right. 

(288) 


NOTES*  289 

ni.— CHAP.  V.    The  general  equation,  Ay' 4.  Bzy -f-  Czs4-  Dy-f  Ez-f-  F=0, 
of  conic  sections  contains  but  five  arbitrary  constants,  since  we  may  divide  all 
its  terms  by  the  coefficient  of  any  one  term.     Therefore  a  conic  section  may 
be  made  to  fulfil  five  distinct  conditions  (such  as  passing  through  five  given 
points,  only  two  of  which  lie  on  the  same  right  line)  provided  none  of  these 
constants  are  determined  by  the  analytical  condition  which  determines  the 
class  of  the  curve.     If  the  curve  be  an  ellipse,  we  must  have,  B2  —  4AC  <  0, 
which  does  not  determine  any  of  the  constants  A,  B,  C,  and  therefore  the 
ellipse  can  be  made  to  pass  through  five  given  points.     Also,  its  most  general 
equation  must  contain  five  arbitrary  constants,  which  are,  either  directly  or 
indirectly,  the  co-ordinates  of  the  centre,  the  lengths  of  the  axes,  and  the 
direction  of  one  of  them.     When  the  ellipse  becomes  a  circle  we  must  have, 
A  =  C,  and  B  =  0,  by  which  two  of  the  constants  are  determined,  leaving  only 
three  arbitary  constants  in  the  equation :  so  that  the  circle  can  be  made  to 
pass  through  but  three  given  points.     If  the  curve  be  a  parabola,  we  must 
have  B2  —  4AC  —  0,  which  determines  one  constant,  thus  leaving  four  in  the 
equation ;  so  that  the  parabola  can  be  made  to  pass  through  but  four  given 
points.     Its  most  general  equation  must  contain  four  independent  constants, 
which  are,  either  directly  or  indirectly,  the  co-ordinates  of  the  vertex,  the 
parameter,   and  the  direction  of  the  axis.     The  student  can  readily  apply 
these  principles  to  the  varieties  of  this  class  of  curves. 

If  the  curve  be  a  hyperbola,  we  must  have,  B7  —  4AC  >  0,  which  deter 
mines  none  of  the  constants,  and  therefore  this  curve  may  be  made  to  pass 
through  five  given  points.  Its  most  general  equation  must  contain  five  arbi 
trary  constants,  the  same  as  for  the  ellipse.  The  equilateral  hyperbola  can  be 
made  to  pass  through  but  three  given  points.  When  the  hyperbola  degenerates 
into  two  straight  lines,  the  roots  z',  zx/,  must  be  equal,  which  can  only  happen 
when  the  quantity  under  the  radical  is  a  perfect  square.  This  requires  that 
the  coefficient  of  the  middle  term  shall  be  equal  to  the  double  product  of  the 
square  roots  of  the  coefficients  of  the  extreme  terms.  The  equation  expressing 
this  condition  determines  one  constant,  thus  leaving  but  four  arbitrary  con 
stants  in  the  equation  of  the  curve ;  so  that  two  straight  lines  which  intersect 
can  be  made  to  pass  through  only /our  points. 

The  close  of  this  discussion  would  seem  to  be  the  proper  place  for  intro 
ducing  some  notice  of  the  origin  of  the  Conic  Sections.  They  were  first  dis 
covered  in  the  school  of  Plato ;  and  his  disciples,  excited,  no  doubt,  by  the 
many  beautiful  properties  of  these  curves,  examined  them  with  such  assiduity, 

25  2M 


290  NOTES. 

that  in  a  very  short  time  several  complete  treatises  on  them  were  published. 
Of  these,  the  best  still  extant  is  that  of  Apollonius  of  Perga,  who  acquired 
from  his  works  the  title  of  the  Great  Geometrician.  His  treatise  on  these 
curves  has  come  down  to  us  only  in  a  mutilated  form,  but  is  well  worth  atten 
tion,  as  showing  how  much  could  be  done  by  the  ancient  analysis,  and  as 
giving  a  very  high  opinion  of  the  geometrical  genius  of  the  age.  Apollonius 
gave  the  names  of  ellipse  and  hyperbola  to  those  curves  —  Hyperbola,  because 
the  square  on  the  ordinate  is  equal  to  a  figure  "  exceeding"  ("wrsp/JocXXoi'")  the 
rectangle  on  the  abscissa  and  parameter. 

Ellipse,  because  the  square  on  the  ordinate  is  "defective"  ("  eAAeiTroi/")  with 
respect  to  the  same  rectangle.  It  is  not  known  who  gave  the  name  of  para 
bola  to  that  curve  —  probably  Archimedes,  because  the  square  on  the  ordinate 
is  equal  (  "  Trocpac/JaAW')  to  this  rectangle. 

Thus,  the  ancients  viewed  these  curves  geometrically,,  in  the  same  manner 
as  we  are  accustomed  to  express  them  by  the  equation,  y*=  mx-\-  nxz. 

IV.  —  Art.  329.     In  the  polar  equation  of  the  conchoid  here  given,  the 
pole  is  supposed  to  be  at  the  point  A  (Fig.  7),  and  the  line  BC  is  the  fixed 
axis  from  which  the  angle  0  is  estimated. 

V.  —  Art.  311.     We  had  designed  leaving  the  proof  of  this  construction  as 
an  exercise  for  the  st-udent,  but  it  may  not,  perhaps,  be  advisable  to  omit 
establishing  the  truth  of  so  important  a  method.     Take  0  (Fig.  a,  page  212) 
as  the  origin,  and  OB,  AO  as  the  axes  of  x  and  y.     Put  OD  =  d,  OB  =  6, 

xy  x       y 

AO  =  a,  OC  -—  c.     The  equation  of  DC  is,  -j  -f  —  =  1  ;  of  AB,  j  -f-  —  =  1  ; 

of  AD,  -|  -f  -|  =  1  ;  of  BC,  -|  -f-  -|  =  1.     Then  that  of  PH  is, 


The  equation  of  the  curve  is, 

Ay'-f  Bzy-f  Cz'+Dy-f  E*+F  =  0  ......  (2). 

To  get  the  points  B  andD,  makey  =  o  in  (2),  which  gives,  Cz2-{-Ea;-f  F  =  0, 
whose  roots  are  the  values  of  6  and  d.  Hence  by  the  theory  of  equations, 

^-4.-=  —  -.     Similarly,  --f-  =  —  p.      Hence   (1)   becomes,    Dy-f 

Ex  4-  2F  =  0,  which  is  the  polar  line  of  the  origin  0.  Similarly  OH  is  the 
polar  line  of  P,  and  PO  that  of  H,  which  renders  the  truth  of  our  construc 
tions  evident. 


APPENDIX. 


I. 

TRIGONOMETRICAL  FORMULA 

N.  B  —  Radius  is  counted  as  1. 

sin  A  . 

1.  Tang  A  =  -  r 

cos  A 

2.  Cot  A  =  -r—  r-  • 

sin  A 

3.  Sec  A  =  ---  T* 

cos  A 

4.  Cosec  A  =  -  —  T— 

sin  A 

5.  Sin  (A  +  B)  =  sin  A  cos  B  -f  sin  B  cos  A. 

6.  Cos  (A  +  B)  =  cos  A  cos  B  —  sin  A  sin  B. 

7.  Sin  (A  —  B)  =  sin  A  cos  B  —  sin  B  cos  A. 

8.  Cos  (A  —  B)  =  cos  A  cos  B  +  sin  A  sin  B 


9. 


10.  Tang  (A  -  B)  = 


B 

—  tang  A  tang  B 

ta"2  A  ~  tan*  B 


1  +  tang  A  tang  B 

11.  Tang  2A  =  -, — ta"g   ,.  • 
1  —  tang'A 

(291) 


292  APPENDIX. 

sin  A  +  sin  B  _  tang  £  (A  +  B) 
•*  Sin  A  —  sin  B  ~  tang  i  (A  —  B) 


1  —  cos  A 
15. 


16.  Sin  2A  =  2  sin  A  cos  A. 


.t. 

1  -f  tang2 

18.  Cos  'A  - 


APPENDIX. 


II. 

QUESTIONS  ON  ANALYTICAL  GEOMETRY. 


CHAPTER  I. 

WHAT  is  Algebra  ?  May  it  be  applied  to  the  solution  of  geometrical  pro 
blems  ?  What  is  necessary  to  such  application  ?  What  is  an  unit  of  measure  ? 
In  comparing  lines,  what  kind  of  unit  is  used  ?  Surfaces  ?  Solids  ?  Would  you 
use  the  same  linear  unit  for  comparing  all  lines  ?  What  are  some  of  the  linear 
units  ?  What  are  some  of  the  units  for  comparing  plane  surfaces  ?  Solids  ? 
How  would  you  compare  two  lines  ?  Suppose  one  contained  the  unit  of  5  times 
and  the  other  10  times,  how  would  they  compare  ?  How  would  you  compare 
surfaces?  If  a  surface  were  represented  by  the  number  10,  what  would  this 
number  express  ?  If  another  were  expressed  by  20,  how  would  the  two  com 
pare  ?  If  the  solidity  of  a  body  be  represented  by  50,  what  would  this  number 
denote  ?  How  then  may  we  conceive  lines,  surfaces,  &c.,  to  be  added  to  each  other? 
May  all  the  operations  of  arithmetic  be  thus  performed  upon  them  ?  How  ?  If 
the  length  of  two  lines  be  expressed  numerically  by  a  and  6,  how  might  the  lines 
be  added  ?  What  would  the  sum  of  the  two  lines  be  equal  to  ?  What  is  meant 
by  the  construction  of  a  geometrical  expression  ?  How  might  you  construct  a 
line  that  should  be  equal  to  the  sum  of  two  given  lines  ?  Their  difference  ?  What 
do  the  numbers  which  represent  the  lines  denote  ?  How  may  you  pass  from 
the  equation  between  the  numerical  values  of  the  lines  to  that  between  their 
absolute  lengths  ?  Will  the  two  sets  of  equations  ever  be  of  the  same  form  ? 
When  ?  Is  it  necessary  in  such  cases  to  make  the  transformation  ?  Why  not  ? 
When  will  not  the  two  sets  of  equations  be  of  the  same  form  ?  May  homoge 
neous  equations  be  at  once  constructed  without  transformation  ?  Would  the 
equation  x  =  ab,  express  a  numerical  or  geometrical  relation  ?  Why  nume 
rical  ?  In  order  that  it  should  express  a  geometrical  relation,  what  must  the  unit 
of  measure  be  denoted  by  ?  How  may  you  construct  an  equation  of  the  form 

x  =  abed?     x  =  \/ab  ?      x  —  ^/a-  4.  II-  ?      x  _=.  v/a b-  ?      When  a  quad. 

•atic  equation  has  to  be  constructed,  what  does  an  imaginary  value  for  x  denote  ? 

25*  (293) 


294  APPENDIX. 

Suppose  the  values  of  x  are  equal  ?  Unequal  ?  What  interpretation  is  given  to 
negative  solutions  ?  Is  this  a  common  interpretation  ?  How  was  the  negative 
solution  interpreted  in  the  problem  of  the  couriers  in  Algebra? 


CHAPTER  II. 

How  is  Analytical  Geometry  divided  ?  What  is  Determinate  Geometry  ?  In. 
determinate  Geometry  ?  Give  an  example  of  the  problems  embraced  in  Deter- 
ruinate  Geometry.  What  are  the  general  steps  to  be  followed  to  express  analy 
tically  the  condition  of  geometrical  problems  ?  How  many  equations  must  there 
be  ?  How  are  the  solutions  obtained  ?  Who  first  applied  Algebra  to  Geometry  ? 
(Vieta.) 

CHAPTER  III. 

What  kind  of  questions  are  embraced  in  Indeterminate  Geometry  ?  Why 
are  such  problems  called  indeterminate  ?  What  does  the  equation  y  —  x  ex 
press  ?  Does  it  define  fully  a  straight  line  ?  Wrhat  does  the  equation  y2  =  2aa; 
—  ar3  denote  ?  Why  the  circumference  of  a  circle?  May  every  line  be  thus 
represented  by  an  equation  ?  May  every  equation  be  interpreted  geometrically  ? 
Who  first  made  this  more  extended  application  of  Algebra  to  Geometry  ? 
(Descartes,} 

How  do  you  define  space  ?  Can  the  absolute  positions  of  bodies  be  determined  ? 
May  their  relative  positions  ?  In  what  manner  ?  How  may  the  relative  posi 
tions  of  points  in  a  plane  be  fixed  ?  What  are  the  assumed  lines  called  ?  What 
is  the  origin  ?  What  is  an  abscissa  ?  An  ordinate  ?  What  is  meant  by  variables  ? 
Constants  ?  When  is  the  position  of  a  point  fixed  ?  What  are  the  equations  of 
a  point  ?  If  the  abscissa  be  constant  while  the  ordinate  varies,  how  will  the 
position  of  the  point  be  effected  ?  If  the  ordinate  be  constant  and  the  abscissa 
vary  ?  What  are  the  equations  of  the  origin  ?  How  are  points  in  the  four 
angles  of  the  co-ordinate  axes  represented?  What  are  the  equations  of  a  point 
in  the  2d  angle  ?  3d  ?  4th  ?  In  the  first  angle  on  the  axis  of  x  ?  on  the  axis  of 
y  ?  In  the  third  angle  on  the  axis  of  x  ?  of  y?  What  does  the  equation  x=a 
considered  alone  denote  ?  y  =  b  ?  How  is  it  then  that  the  two  combined  fix  the 
position  of  a  point  in  a  plane  ?  What  does  the  equation  of  a  line  express  ?  Why 
is  the  equation  of  a  straight  line  in  a  plane  referred  to  oblique  axes  ?  How  do 
you  know  it  is  the  equation  of  a  straight  line  ?  May  this  equation  express  a 
straight  line  in  every  position  it  may  take  in  tho  plane  of  the  axes  ?  Suppose  it 
pass  through  the  origin  ?  If  it  cut  the  axis  of  ordinates  above  the  origin  ?  below  ? 
How  is  it  situated  if  the  co-efficient  of  x  be  negative  ?  How  is  the  point  deter 
mined  in  which  it  cuts  the  axis  of  x  ?  of  y?  What  is  the  equation  of  a  right 
line  referred  to  rectangular  axes  ?  What  is  the  reason  of  the  change  ?  What 
does  the  co-efficient  of  the  variable  in  the  second  member  express  ?  The  abso 
lute  term?  What  will  be  its  equation  if  it  be  parallel  to  the  axis  ofy?  If  it  be 
parallel  to  the  axis  of  x  ?  If  it  pass  through  the  origin  ?  Which  of  the  quan* 


APPENDIX.  295 

titles  in  the  equation  of  a  straight  line  referred  to  rectangular  axes  fixes  its  posi 
tion?  Must  a  and  b  be  both  known  to  determine  the  line  ?  If  a  be  known,  and  b 
be  indeterminate,  what  will  the  equation  denote  ?  If  b  be  known,  and  a  inde 
terminate  ?  If  both  a  and  b  be  indeterminate  ?  How  many  separate  conditions 
may  a  straight  line  be  made  to  fulfil  ?  What  is  the  equation  of  a  straight  line 
passing  through  a  given  point  ?  Why  must  a  or  b  disappear  in  the  process 
for  obtaining  this  equation  ?  What  is  the  equation  of  a  straight  line  passing 
through  two  given  points  ?  Why  do  a  and  b  both  form  this  equation  ?  If  the 
given  points  have  the  same  abscissa,  what  will  the  equation  of  the  line  become  ? 
If  they  have  the  same  ordinate  ?  What  is  the  condition  for  two  parallel  straight 
lines  ?  \Vhat  is  the  expression  for  the  tangent  of  the  angle  which  two  straight 
lines  make  with  each  other  in  a  plane  ?  What  is  the  condition  of  two  perpen 
dicular  straight  lines  in  a  plane  ?  How  do  you  ascertain  the  point  of  intersec 
tion  of  two  straight  lines  in  a  plane  ?  How  is  the  distance  between  two  points 
in  a  plane  expressed  ?  If  one  of  the  points  be  the  origin  ? 

Of  Points  and  Line  in  Space. 

How  is  a  point  in  space  determined  ?  What  are  the  planes  used  called  ?  What 
are  the  co-ordinate  axes  ?  What  are  the  co-ordinates  of  a  point  in  space  ?  How 
are  they  measured  ?  What  is  the  origin  ?  What  are  the  equations  of  a  point 
in  space  ?  What  is  meant  by  the  projection  of  a  point  ?  How  many  projections 
will  a  point  have  ?  What  are  the  equations  of  the  projection  of  a  point  on  the 
plane  ofxy?  xz  ?  yz?  If  the  projections  of  a  point  on  the  planes  xy  and  xz 
were  known,  could  you  determine  the  equations  of  the  third  projection  ?  How  ? 
Could  you  make  the  geometrical  construction  for  the  third  projection  ?  How  ? 
If  one  of  the  equations  of  a  point  in  space,  as  x  =  a,  be  considered  by  itself, 
what  does  it  express  ?  What  does  the  equation  y  =  b  represent  1  z  =  c  ?  If 
two  of  these  equations  be  considered  together,  what  would  they  represent  with 
reference  to  the  position  of  the  point  ?  Would  they  be  sufficient  to  define  it  ? 
If  the  third  equation  be  connected  with  the  other  two,  would  the  three  be  suffi 
cient  ?  Why  ?  What  are  the  equations  of  the  origin  ?  What  are  the  equations 
of  a  point  on  the  axis  of  x  ?  of  y  ?  of  z?  What  signification  have  negative  co 
ordinates  ?  What  is  the  expression  for  the  distance  between  two  points  in  space  ? 
If  one  be  the  origin  of  co-ordinates  ?  To  what  is  the  square  of  the  diagonal  of 
a  parallelopipedon  equal  ?  To  what  is  the  sum  of  the  squares  of  the  cosines  of 
the  angles  which  a  straight  line  in  space  makes  with  the  co-ordinate  axes  equal  ? 
How  are  the  equations  of  a  straight  line  in  space  determined  ?  What  are  they? 
What  do  they  represent  ?  Knowing  the  equations  of  two  projections  of  a  line, 
may  you  determine  the  equation  of  the  third  projection  ?  What  is  meant  by  the 
projection  of  a  line  ?  How  many  equations  are  necessary  to  fix  the  position  of 
a  straight  line  in  space  ?  Why  only  two  ?  What  quantities  in  these  equations 
fix  its  position  ?  When  the  constants  are  arbitrary,  what  is  the  position  of  the 
line  ?  Do  you  know  it  is  a  straight  line  ?  Suppose  one  of  the  constants  ceases 
"  to  be  arbitrary,  what  effect  upon  the  position  of  the  line  ?  If  two  ?  If  all  are 
known?  What  is  the  projection  of  a  curve?  How  mav  its  position  be  fixed 


296  APPENDIX. 

analytically  ?  What  are  the  equations  of  a  line  passing  through  two  points  in 
space  ?  What  is  the  expression  for  the  cosine  of  the  angle  between  two  lines  in 
space,  in  terms  of  the  angles  which  they  make  with  the  co-ordinate  axes  ?  In 
terms  of  their  constants  ?  What  is  the  condition  of  perpendicularity  of  two 
lines  in  space  ?  Of  parallelism  ?  How  do  you  determine  the  intersection  of 
two  lines  ?  How  is  the  condition  that  the  lines  shall  intersect  expressed  ? 

Of  the  Plane. 

How  do  you  define  a  plane  ?  How  is  the  equation  of  a  plane  determined,  if 
it  be  regarded  as  the  locus  of  perpendiculars  ?  Why  do  you  eliminate  the  con. 
stants  in  the  equations  of  the  perpendiculars  ?  Why  will  the  resulting  equation 
be  that  of  a  plane  ?  What  are  the  traces  of  a  plane  ?  How  are  the  equations 
of  the  traces  determined  ?  If  a  line  be  perpendicular  to  a  plane  in  space,  how 
will  the  projections  of  the  line  be  situated  ?  What  is  the  most  general  equation 
of  the  first  degree  between  three  variables  ?  What  does  it  represent  ?  Why  a 
plane  ?  If  the  plane  be  perpendicular  to  ary,  what  will  be  its  equation  ?  To 
xz  1  to  yz  ?  What  is  the  equation  of  the  plane  xz  1  xy  ?  yz  ?  Of  a  plane  pa- 
rallel  to  xy  ?  to  xz  ?  to  yz  ?  Of  a  plane  passing  through  the  origin  ?  How  do 
you  determine  the  equation  of  a  plane  passing  through  three  given  points  ?  Is 
this  problem  always  determinate  ?  Why  ?  How  do  you  determine  the  equations 
of  the  intersections  of  two  planes  ?  If  you  eliminate  one  of  the  variables,  what 
does  the  resulting  equation  express  ? 

Transformation  of  Co-ordinates. 

How  are  curves  divided  ?  What  are  Algebraic  curves  ?  Transcendental 
curves  ?  Give  an  example  of  each.  How  are  Algebraic  curves  classified  ? 
What  order  is  the  equation  of  a  straight  line  ?  WThat  is  meant  by  the  discussion 
of  a  curve  ?  How  may  this  discussion  be  oftentimes  simplified  ?  Do  the  trans 
formations  of  co-ordinates  affect  the  character  of  the  curve  ?  In  what  do  they 
consist?  How  is  the  transformation  effected  ?•  What  are  the  equations  of  trans 
formation  from  one  system  of  rectangular  axes  to  a  parallel  system  ?  To  an 
oblique,  the  origin  remaining  the  same  ?  From  oblique  to  oblique  ?  In  what 
kind  of  functions  is  the  relation  between  the  old  and  new  co-ordinates  expressed  ? 
Is  the  relation  linear  if  the  transformation  be  made  in  space  ?  How  many 
equations  for  transformation  in  space  ?  What  does  each  set  of  equations  ex 
press  ?  If  the  new  axes  be  rectangular,  what  condition  in  their  equations  does 
it  require  ?  What  are  polar  co-ordinates  ?  What  is  the  pole  ?  Radius  vector  ? 
What  are  the  polar  co-ordinates  when  the  origin  is  not  changed  ?  When  it  is  ? 
If  the  axis  from  which  the  variable  angle  is  estimated  is  not  parallel  to  x  ?  What 
do  negative  values  of  the  radius  vector  indicate  ?  Why  ? 

CHAPTER  IV. 

Conic  Sections. 

What  are  the  Conic  Sections  ?  How  is  a  right  line  generated  ?  How  may 
its  equation  be  determined  ?  What  is  its  form  ?  How  may  the  general  equation 


APPENDIX.  297 

of  intersection  of  a  cone  and  plane  be  determined  ?  How  many  different  forms 
of  curves  result  from  the  intersection  ?  What  changes  are  made  on  the  general 
equation  cf:r.*.:r::cticn,  to  deduce  the  equations  of  these  separate  curves?  What 
is  the  general  character  of  the  curves  called  Ellipses  ?  Parabolas  ?  Hyper- 
bolas  ?  What  is  the  direction  of  the  cutting  plane  to  produce  ellipses  ?  Para 
bola  ?  Hyperbola  ?  Circle  ?  What  distinguishes  the  equation  of  the  ellipse 
from  that  of  the  hyperbola  ?  Parabola  ?  If  the  cutting  plane  pass  through  the 
>ertex,  what  do  the  ellipse  and  circle  become  ?  Parabola?  Hyperbola?  How 
qre  these  results  proved  by  the  equations  of  these  curves  ? 

Of  ike  Circle. 

How  is  the  circle  cut  from  the  cone  ?  What  is  the  form  of  its  equation  ? 
What  property  results  from  the  form  of  its  equation  ?  How  do  you  determine 
the  points  in  which  the  curve  cuts  the  axis  of  x  ?  ofy?  How  do  their  distances 
from  the  centre  compare  ?  How  do  you  determine  intermediate  points  ?  When 
do  real  values  for  y  result  ?  When  imaginary  ?  What  relation  between  the 
ordinate  of  any  point  of  the  circumference,  and  the  divided  segments  of  the 
diameter  ?  What  are  supplementary  chords  ?  How  are  they  related  in  the 
circle  ?  What  is  the  equation  of  the  circle  referred  to  the  extremity  of  a  diameter  ? 
To  axes  without  the  circle  ?  How  is  the  equation  of  a  tangent  line  determined  ? 
What  is  its  form  ?  Of  a  normal  line  ?  Through  what  point  do  all  the  normal 
lines  of  the  circle  pass?  What  are  conjugate  diameters  ?  Has  the  circle  conju. 
gate  diameters  ?  How  many  ?  In  what  position  ?  How  do  you  determine  the 
polar  equation  of  the  circle  ?  How  is  this  equation  made  to  express  all  the  points 
of  the  curve  ?  Suppose  the  pole  is  on  the  circumference  ?  At  the  centre  ? 

Of  the  Ellipse. 

What  direction  has  the  cutting  plane  when  the  conic  section  is  an  ellipse  ? 
What  is  the  form  of  its  equation  ?  How  do  you  discuss  this  equation  ?  What 
is  the  equation  of  the  ellipse  referred  to  its  centre  and  axes  ?  What  do  A  and 
B  express  in  this  equation  ?  What  is  the  longest  diameter  in  the  ellipse  called  ? 
Shortest  ?  If  its  axes  be  equal,  what  does  the  equation  become  ?  What  is  a 
diameter  ?  a  parameter  ?  What  relation  between  the  ordinates  of  the  curves 
and  the  corresponding  segments  of  the  diameter  ?  If  two  circles  be  described 
upon  the  axes,  what  relation  will  they  bear  to  the  ellipse  ?  What  relation  will 
exist  between  their  ordinates  ?  How  may  this  property  enable  you  to  describe 
the  ellipse  by  points  ?  What  relation  do  the  supplementary  chords  in  the  ellipse 
bear  to  each  other  ?  What  are  the  foci  of  the  ellipse  ?  What  properties  do 
these  points  possess  ?  What  is  the  eccentricity  ?  What  is  its  maximum  value  ? 
Minimum  ?  What  does  the  ellipse  reduce  to  in  the  first  case  ?  In  the  second  ? 
What  are  the  various  modes  of  describing  the  ellipse  ?  What  is  the  equation 
of  a  tangent  line  to  the  ellipse  ?  Normal  ?  What  relation  exists  between  the 
angles  which  the  tangent  line  makes  with  the  axis  of  x,  and  those  which  the 
supplementary  chords  make?  How  may  you  draw  a  tangent  line  by  this  pro 

2N 


298  APPENDIX. 

perty  ?  What  is  a  subtangent  ?  What  is  its  value  in  the  ellipse  ?  Knowing 
the  subtangent,  how  may  a  tangent  line  be  drawn  ?  What  is  the  normal  ?  What 
relation  between  the  tangent  and  normal  ?  How  does  this  relation  enable  you 
to  draw  a  tangent  line  ?  Has  the  ellipse  conjugate  diameters  ?  How  many  ? 
How  many  are  perpendicular  to  each  other  ?  What  is  the  rectangle  upon  the 
axes  equal  to  ?  Sum  of  the  squares  of  the  axes  ?  How  may  you  draw  two 
conjugate  diameters,  making  a  given  angle  with  each  other?  How  may  the 
polar  equation  define  the  curve?  Suppose  the  pole  at  the  centre?  At  one  of 
the  foci  ?  Upon  the  curve  ?  When  the  radius  vector  is  negative,  what  does  it 
signify  ?  May  you  determine  the  equation  of  the  ellipse  from  one  of  its  pro 
perties  ?  Illustrate  this.  What  is  the  area  of  the  ellipse  equal  to  ?  How  do 
the  areas  of  two  ellipses  compare  ? 

Of  the  Parabola. 

What  is  the  direction  of  the  cutting  plane  when  the  conic  section  is  a  parabola  ? 
Its  equation  ?  How  do  you  discuss  this  equation  ?  Its  parameter  ?  How  do 
the  squares  of  the  ordinates  compare  ?  How  is  the  curve  described  7  Its  focus  ? 
Direction  ?  What  relation  between  the  two  ?  What  method  of  describing  the 
parabola  results  ?  What  is  the  double  ordinate  through  the  focus  equal  to  ? 
Equation  of  tangent  line  ?  To  what  is  the  subtangent  equal  ?  Subnormal  ?  What 
relation  between  tangent  and  normal  ?  How  may  you  draw  a  tangent  line  to 
a  parabola  ?  Has  this  curve  diameters  ?  How  situated  ?  What  is  the  position 
of  a  new  system  of  axes,  that  the  curves  shall  preserve  the  same  form  when 
referred  to  them  ?  What  is  the  polar  equation  ?  How  does  it  define  the  curve  ? 
If  the  pole  be  at  the  focus  ?  On  the  curve  ?  May  you  deduce  the  equation  of 
the  curve  from  one  of  its  properties  ?  Illustrate.  What  is  the  measure  of  any 
portion  of  the  parabola?  What  are  quadrable  curves  ?  Is  this  curve  quadrable? 

Of  the  Hyperlola. 

What  direction  has  the  cutting  plane  when  the  conic  section  is  an  hyperbola  ? 
What  is  the  form  of  its  equation  ?  How  is  it  distinguished  from  the  ellipse  ? 
How  do  you  discuss  this  equation  ?  What  is  the  equation  referred  to  the  centre 
and  axes  ?  Equilateral  hyperbola  ?  What  relation  between  supplementary 
chords  ?  What  is  the  conjugate  hyperbola  ?  What  are  the  foci  of  this  curve  ? 
What  properties  do  they  possess  ?  How  is  the  curve  constructed  ?  What  is  the 
equation  of  its  tangent  line  ?  What  relation  between  the  tangent  lines  and  sup 
plementary  chords  ?  How  may  you  draw  a  tangent  line  to  the  curve  ?  Has 
the  hyperbola  conjugate  diameters  ?  To  what  is  the  difference  of  the  squares 
on  the  conjugate  diameters  equal  ?  How  are  the  conjugate  diameters  of  the 
equilateral  hyperbola  related  ?  What  is  the  rectangle  on  the  axes  equal  to  ? 
What  are  the  asymptotes  of  this  curve  ?  What  is  their  equation  ?  What  lines 
do  they  limit  ?  How  may  you  construct  them  ?  What  is  the  form  of  the  equa. 
non  of  the  hyperbola  referred  to  them  ?  What  is  the  power  of  the  hyperbola  ? 
When  the  hyperbola  is  equilateral,  what  does  the  equation  referred  to  its  asymp 


APPENDIX.  299 

totes  become?  How  is  a  tangent  line  to  the  hyperbola  divided  at  the  point  of 
tangency  ?  If  any  line  be  drawn,  intersecting  the  hyperbola  and  limited  by  the 
asymptotes,  what  property  exists  ?  How  does  this  property  enable  you  to  con 
struct  points  of  the  curve?  What  is  the  polar  equation  of  this  curve?  How 
does  it  define  the  curve  ?  If  the  pole  be  at  the  centre  ?  At  one  of  the  foci  ? 
Upon  the  curve  ?  May  the  same  polar  equation  represent  each  of  the  conie 
sections?  In  what  manner  may  you  pass  from  one  to  the  other ?  Mention  the 
distinctive  characteristics  in  the  forms  of  the  conic  sections.  Mention  their 
common  properties.  Their  analogies. 


CHAPTER  V. 
Discussion  of  Equations 

What  is  the  most  general  form  of  an  equation  of  the  2d  degree  with  two 
variables  ?  Give  an  analysis  of  the  mode  of  discussing  it.  Why  may  you 
omit  in  the  general  discussion  the  case  in  which  the  squares  of  the  variables 
are  wanting?  How  are  the  curves  represented  by  this  equation  classified? 
What  suggests  this  mode  of  classification?  What  is  the  analytical  character 
of  curves  of  the  1st  class?  2d  class?  3d  class?  How  do  you  discuss  the 
1st  class?  What  results  from  the  discussion?  How  is  the  limited  nature  of 
the  curves  apparent?  How  apply  the  principles  to  a  numerical  example? 
How  determine  to  which  class  of  curves  a  particular  equation  belongs  ?  What 
are  the  particular  cases  comprehended  in  the  first  class  ?  In  the  case  in  which 
A  =  C,  and  B  =  0,  what  does  the  equation  represent  if  the  co-ordinate  axes 
be  obli-que?  (Ans.  An  ellipse  referred  to  its  equal  conjugate  diameters.)  How 
do  you  discuss  the  2d  class?  What  part  of  the  equation  represents  the 
diameter  of  these  curves?  What  are  the  varieties  of  this  class  ?  What  curves 
do  they  resemble?  How  do  you  discuss  the  3d  class ?  What  varieties?  What 
curves  do  they  resemble  ?  What  is  the  centre  of  a  curve  ?  Its  diameter  ? 
What  conditions  must  the  equation  of  a  curve  fulfil  when  referred  to  its  centre? 
Have  curves  of  the  2d  order  centres  ?  Which  of  them  ?  How  many  ?  Why 
only  one?  In  which  class  are  the  conditions  for  a  centre  impossible?  Why? 
What  conditions  must  the  equation  of  a  curve  fulfil  when  referred  to  a  diam 
eter?  If  both  co-ordinate  axes  are  diameters  ?  If  axis  of  y?  Ifxf  Which 
of  the  curves  of  the  2d  order  have  diameters  ?  How  are  they  situated  in  the 
2d  class?  Have  any  of  these  curves  asymptotes?  Which?  Why  only  those 
of  the  3d  class?  How  can  you  find  the  asymptotes  from  the  equation  of  the 
curve  ?  Do  these  properties  show  much  resemblance  between  these  curves  and 
the  conic  sections  ?  How  far  does  the  resemblance  extend  ?  How  is  the  per 
fect  identity  proved  ?  Then  every  equation  of  the  2d  degree,  with  two  varia 
bles,  must  represent  what?  When  an  ellipse ?  Parabola?  Hyperbola?  How- 
many  conic  sections  are  there,  including  the  varieties  ?  Through  how  many 
points  may  an  ellipse  be  made  to  pass  ?  A  parabola  ?  Hyperbola?  Equilateral 


300  APPENDIX. 

hyperbola?  How  many  constants  must  the  most  general  equation  of  the 
ellipse  contain  ?  What  are  they  ?  How  many  must  be  contained  by  that  of 
the  parabola  ?  What  are  they  ?  How  many  by  that  of  the  hyperbola  ?  How 
many  by  that  of  the  equilateral  hyperbola  ?  What  are  they  ?  If  the  curve 
be  an  ellipse,  will  the  terms  involving  z*  and  y2  have  the  same  or  different 
signs?  How  is  it  with  the  parabola?  How  with  the  hyperbola?  If,  in  the 
general  equation  of  the  2d  degree  with  two  variables,  the  term  involving  the 
rectangle  of  the  variables  be  wanting,  what  must  you  infer?  (Ans.  That  the 
curve  is  referred  to  co-ordinate  axes  parallel  to  a  diameter  and  the  tangent  at 
its  extremity.)  Why  ?  The  presence  of  the  term  Exy  in  the  equation  is  due 
to  what?  (Ans.  To  the  directions  of  the  co-ordinate  axes.)  What  if  the  ab 
solute  term  be  wanting  ?  What  if  the  terms  containing  the  first  powers  of  the 
variables  be  absent?  (Ans.  That  the  origin  is  at  the  centre.)  The  presence 
of  the  terms  Dy,  Ex,  is  then  due  to  what  ?  (Ans.  To  the  removal  of  the  origin 
from  the  centre.)  What  is  the  most  general  equation  of  a  tangent  line  to  a 
conic  section  ?  How  do  you  find  this  equation  ?  By  its  aid  what  remarkable 
property  of  these  curves  is  demonstrated  ?  What  is  a  polar  line  ?  A  pole  ? 
How  would  you  construct  the  polar  line  of  a  given  pole  ?  How  the  pole  of  a 
given  polar  line  ?  How  use  them  for  drawing  a  tangent  line  to  a  conic  section 
from  a  given  point  without  the  curve  ?  How  to  draw  a  tangent  from  a  given 
point  upon  the  curve  ?  What  is  the  peculiar  advantage  of  these  methods  ? 
(Ans.  That  we  can  draw  the  tangent  without  knowing  the  species  of  the  sec 
tion.)  In  the  parabola,  what  point  is  the  pole  of  the  directrix?  Tangents 
which  intersect  upon  the  directrix  make  what  angle  with  each  other9 


CHAPTER  VI. 

Curves  of  the  Higher  Orders. 

What  is  the  objection  to  attempting  a  systematic  examination  of  curves? 
What  is  the  3d  order  remarkable  for  ?  How  many  curves  does  this  order  com 
prise  ?  How  many  of  them  were  discussed  by  Sir  I.  Newton  ?  What  is  the 
number  of  varieties  included  in  the  4th  order  ?  Is  a  complete  investigation 
of  curves  necessary  ?  Why  not  ?  Give  an  outline  of  the  general  method  to 
be  pursued  in  determining  the  form  of  any  curve  from  its  equation.  How  is 
the  cissoid  generated  ?  Its  equation?  Its  polar  equation  ?  By  whom  invented  ? 
For  what  purpose  ?  Whence  its  name  ?  Has  it  an  asymptote  ?  Explain  the 
generation  of  the  conchoid.  Its  equation.  Its  polar  equation.  How  are  the 
two  parts  distinguished  ?  Are  they  both  defined  by  one  equation  ?  What  is 
the  modulus  ?  The  base,  or  rule  ?  How  many  cases  may  you  distinguish  in 
its  discussion  ?  What  are  they  ?  What  remarkable  point  occurs  in  the  3d 
case  ?  Has  the  curve  an  asymptote  ?  By  whom  was  it  invented  ?  For  what 
purpose  ?  Whence  its  name  ?  How  may  it  be  applied  to  trisecting  an  angle  ? 
How  may  you  solve  the  celebrated  problem  of  the  duplication  of  the  cube  by 


APPENDIX.  301 

conic  sections?  What  is  the  polar  equation  of  the  Lemniscata  of  Bernouilli? 
This  curve  is  the  locus  of  what  series  of  points  ?  What  is  its  form  ?  What 
remarkable  property  does  it  possess?  What  are  Parabolas  of  the  higher 
orders?  Their  general  equation ?  What  varieties  are  noticed?  The  equa 
tion  of  the  semi-cubical  parabola?  From  what  does  it  take  its  name?  Its 
polar  equation  ?  For  what  is  it  remarkable  ?  Form  of  the  curve  ?  Equation 
of  the  cubical  parabola  f  Its  polar  equation  ?  Form  of  the  curve  ?  What  are 
transcendental  curves?  Whence  the  name?  What  is  the  Logarithmic  curve? 
Its  equation  ?  What  is  the  axis  of  numbers  ?  Of  logarithms  ?  By  whom  was 
thi?  curve  invented?  What  are  some  of  its  properties?  How  is  the  cycloid 
generated?  Whence  its  name?  What  is  the  base?  Axis?  Vertex?  What 
is  its  equation  referred  to  the  axis  and  tangent  at  the  vertex  ?  Referred  to 
the  base  and  tangent  at  the  cusp  ?  By  whom  was  this  curve  first  examined  ? 
For  what  is  it  remarkable  ?  Mention  some  of  its  properties.  What  peculiar 
appellations  does  it  derive  in  consequence  of  two  of  them  ?  What  is  the  tro- 
choid?  Its  equation?  What  is  the  curtate  cycloid?  Its  equation?  How  may 
the  class  of  cycloids  be  extended?  What  is  the  Epitrochoid ?  Epicycloid  f 
Hypotrochoid  f  Hypocycloid?  How  obtain  their  equations  ?  What  are  they? 
When  may  the  necessary  elimination  be  effected?  Is  the  number  of  convolu 
tions  limited  ?  What  is  the  cardioide?  Its  polnr  equation  ?  When  does  the 
hypocycloid  become  a  right  line?  The  same  supposition  reduces  the  hypo- 
trochoid  to  what  ?  What  are  spirals  ?  By  whom  invented  ?  For  what  pur 
pose?  What  are  the  chief  varieties?  How  is  the  spiral  of  Archimedes 
generated  ?  What  is  its  equation  ?  What  is  the  pole,  or  eye  of  a  spiral  ? 
What  is  the  general  equation  of  spirals?  To  what  co-ordinates  are  these 
curves  referred?  Equation  of  the  hyperbolic  spiral?  Whence  its  name? 
Has  it  an  asymptote  ?  How  is  the  parabolic  spiral  generated  ?  Its  equation  ? 
Equation  of  the  Logarithmic  spiral?  Does  it  ever  reach  the  pole?  (This 
curve  is  also  known  as  the  equiangular  spiral,  from  the  fact  that  the  angle 
formed  by  the  radius  vector  and  tangent  is  constant :  the  tangent  of  this 
angle  being  equal  to  the  modulus  of  the  system  of  logarithms  used.)  What 
are  the  formulas  for  transition  from  polar  to  rectangular  co-ordinates?  May 
the  polar  equation  of  a  curve  sometimes  be  used  to  advantage  ?  When  ? 
Give  an  example. 


CHAPTER  VIL 
Surfaces  of  the  Second  Order. 

How  are  surfaces  divided?  General  equation  of  surfaces  of  the  2d  order? 
How  may  they  be  discussed  ?  Which  is  the  best  mode  ?  Illustrate  this  method. 
How  should  the  secant  planes  be  drawn  ?  What  preliminary  steps  are  neces 
sary  before  discussing  these  surfaces?  How  are  these  surfaces  divided? 
What  is  the  form  of  the  equation  of  surfaces  which  have  a  centre?  No 

26 


302  APPENDIX. 

centre  ?  May  both  classes  be  represented  by  a  common  equation  ?  What 
conditions  will  give  one  class  and  the  other  ?  How  many  cases  of  surfaces 
which  have  a  centre?  What  does  the  1st  case  embrace?  What  are  the  prin" 
cipal  sections  ?  How  do  you  know  they  represent  ellipsoids  ?  What  varieties  ? 
What  is  the  equation  of  a  sphere?  What  conditions  give  a  cylinder?  Right 
cylinder?  Ellipsoid  of  revolution?  What  does  the  2d  case  embrace  ?  What 
are  hyperboloids  ?  Hyperboloids  of  revolution  ?  What  relation  to  cones  ? 
How  many  cases  of  surfaces  of  no  centre?  1st  case  ?  2d  case?  How  may 
we  draw  a  tangent  plane  to  a  surface  ?  What  is  the  mode  in  surfaces  of  the 
2d  order?  General  form  of  the  equation?  When  drawn  to  surfaces  which 
have  a  centre? 


RETURN  Astronomy/Mathematics/Statistics  Library 
TO— ^   100  Evans  Hall  642-0370 


LOAN  PERIOD  1 
1  MONTH 

2 

3 

4 

5 

6 

ALL  BOOKS  MAY  BE  RECALLED  AFTER  7  DAYS 

DUE  AS  STAMPED  BELOW 

JAM,:U?nnfl 

•JflN  1  8  2000 

FORM  NO.  DD  19 


UNIVERSITY  OF  CALIFORNIA,  BERKELEY 
BERKELEY,  CA  94720 


. 

BTAf. 


